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BEGIN:VEVENT
SUMMARY:Tiago Jardim Da Fonseca (Oxford)
DTSTART:20200422T150000Z
DTEND:20200422T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/1
DESCRIPTION:Title: On Fourier coefficients of Poincaré series\nby Tiago Jardim Da Fo
nseca (Oxford) as part of London number theory seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ila Varma (Toronto)
DTSTART:20200429T150000Z
DTEND:20200429T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/2
DESCRIPTION:Title: Malle's conjecture for octic D4-fields\nby Ila Varma (Toronto) as
part of London number theory seminar\n\n\nAbstract\nWe consider the family
of normal octic fields with Galois group $D_4$\, ordered by their discrim
inants. In forthcoming joint work with Arul Shankar\, we verify the strong
form of Malle's conjecture for this family of number fields\, obtaining t
he order of growth as well as the constant of proportionality. In this tal
k\, we will discuss and review the combination of techniques from analytic
number theory and geometry-of-numbers methods used to prove this and rela
ted results.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Lazda (Warwick)
DTSTART:20200506T150000Z
DTEND:20200506T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/3
DESCRIPTION:Title: A Néron-Ogg-Shafarevich criterion for K3 surfaces\nby Chris Lazda
(Warwick) as part of London number theory seminar\n\n\nAbstract\nThe naiv
e analogue of the Néron–Ogg–Shafarevich criterion fails for K3 surfac
es\, that is\, there exist K3 surfaces over Henselian\, discretely valued
fields K\, with unramified etale cohomology groups\, but which do not admi
t good reduction over K. Assuming potential semi-stable reduction\, I will
show how to correct this by proving that a K3 surface has good reduction
if and only if its second cohomology is unramified\, and the associated Ga
lois representation over the residue field coincides with the second cohom
ology of a certain “canonical reduction” of X. This is joint work with
B. Chiarellotto and C. Liedtke.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chantal David (Concordia)
DTSTART:20200513T150000Z
DTEND:20200513T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/4
DESCRIPTION:Title: Non-vanishing cubic Dirichlet L-functions at s = 1/2\nby Chantal D
avid (Concordia) as part of London number theory seminar\n\nAbstract: TBA\
n
LOCATION:https://stable.researchseminars.org/talk/LNTS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rong Zhou (Imperial)
DTSTART:20200520T150000Z
DTEND:20200520T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/5
DESCRIPTION:Title: Independence of $l$ for Frobenius conjugacy classes attached to abelia
n varieties\nby Rong Zhou (Imperial) as part of London number theory s
eminar\n\n\nAbstract\nLet $A$ be an abelian variety over a number field $E
\\subset \\mathbb{C}$ and let $v$ be a place of good reduction lying over
a prime $p$. For a prime $l\\neq p$\, a theorem of Deligne implies that up
on making a finite extension of $E$\, the Galois representation on the $l$
-adic Tate module factors as $\\rho_l:\\Gamma_E\\rightarrow G_A(\\mathbb{Q
}_l)$\, where $G_A$ is the Mumford-Tate group of $A$. We prove that the co
njugacy class of $\\rho_l(Frob_v)$ is defined over $\\mathbb{Q}$ and inde
pendent of $l$. This is joint work with Mark Kisin.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Tamiozzo (Imperial)
DTSTART:20200527T150000Z
DTEND:20200527T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/6
DESCRIPTION:Title: Bloch-Kato special value formulas for Hilbert modular forms\nby Ma
tteo Tamiozzo (Imperial) as part of London number theory seminar\n\n\nAbst
ract\nThe Bloch-Kato conjectures predict a relation between arithmetic inv
ariants of a motive and special values of the associated $L$-function. We
will outline a proof of (the $p$-part of) one inequality in the relevant s
pecial value formula for Hilbert modular forms of parallel weight two\, in
analytic rank at most one.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (Paris-Saclay)
DTSTART:20200603T150000Z
DTEND:20200603T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/7
DESCRIPTION:Title: Picard ranks of reductions of K3 surfaces over global fields\nby Y
unqing Tang (Paris-Saclay) as part of London number theory seminar\n\n\nAb
stract\nFor a K3 surface $X$ over a number field with potentially good red
uction everywhere\, we prove that there are infinitely many primes modulo
which the reduction of $X$ has larger geometric Picard rank than that of t
he generic fiber $X$. A similar statement still holds true for ordinary K3
surfaces with potentially good reduction everywhere over global function
fields. In this talk\, I will present the proofs via the (arithmetic) inte
rsection theory on good integral models (and its special fibers) of $\\mat
hrm{GSpin}$ Shimura varieties. These results are generalizations of the wo
rk of Charles on exceptional isogenies between reductions of a pair of ell
iptic curves. This talk is based on joint work with Ananth Shankar\, Arul
Shankar\, and Salim Tayou and with Davesh Maulik and Ananth Shankar.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (Toronto)
DTSTART:20200612T150000Z
DTEND:20200612T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/8
DESCRIPTION:Title: p-adic Eisenstein series\, arithmetic holonomicity criteria\, and irra
tionality of the 2-adic $\\zeta(5)$\nby Vesselin Dimitrov (Toronto) as
part of London number theory seminar\n\n\nAbstract\nIn this exposition of
a joint work in progress with Frank Calegari and Yunqing Tang\, I will ex
plain a new arithmetic criterion for a formal function to be holonomic\, a
nd how it revives an approach to the arithmetic nature of special values o
f L-functions. The new consequence to be proved in this talk is the irrati
onality of the 2-adic version of $\\zeta(5)$ (of Kubota-Leopoldt). But I w
ill also draw a parallel to a work of Zudilin\, and try to leave some addi
tional open ends where the holonomicity theorem could be useful. The ingre
dients of the irrationality proof are Calegari's p-adic counterpart of the
Apery-Beukers method\, which is based on the theory of overconvergent p-a
dic modular forms (IMRN\, 2005) taking its key input from Buzzard's theore
m on p-adic analytic continuation (JAMS\, 2002)\, and a Diophantine approx
imation method of Andre enhanced to a power of the modular curve $X_0(2)$.
The overall argument\, as we shall discuss\, turns out to bear a surprisi
ng affinity to a recent solution of the Schinzel-Zassenhaus conjecture on
the orbits of Galois around the unit circle.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yifeng Liu (Yale)
DTSTART:20200617T130000Z
DTEND:20200617T140000Z
DTSTAMP:20260404T111112Z
UID:LNTS/9
DESCRIPTION:Title: Beilinson-Bloch conjecture and arithmetic inner product formula\nb
y Yifeng Liu (Yale) as part of London number theory seminar\n\n\nAbstract\
nIn this talk\, we study the Chow group of the motive associated to a temp
ered global $L$-packet $\\pi$ of unitary groups of even rank with respect
to a CM extension\, whose global root number is $-1$. We show that\, under
some restrictions on the ramification of $\\pi$\, if the central derivati
ve $L'(1/2\,\\pi)$ is nonvanishing\, then the $\\pi$-nearly isotypic local
ization of the Chow group of a certain unitary Shimura variety over its re
flex field does not vanish. This proves part of the Beilinson--Bloch conje
cture for Chow groups and L-functions (which generalizes the B-SD conjectu
re). Moreover\, assuming the modularity of Kudla's generating functions of
special cycles\, we explicitly construct elements in a certain $\\pi$-nea
rly isotypic subspace of the Chow group by arithmetic theta lifting\, and
compute their heights in terms of the central derivative $L'(1/2\,\\pi)$ a
nd local doubling zeta integrals. This confirms the conjectural arithmetic
inner product formula proposed by me a decade ago. This is a joint work w
ith Chao Li.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART:20200708T150000Z
DTEND:20200708T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/12
DESCRIPTION:Title: Partial Frobenius structures\, Tate’s conjecture\, and BSD over fun
ction fields.\nby Jared Weinstein (Boston University) as part of Londo
n number theory seminar\n\n\nAbstract\nTate’s conjecture predicts that G
alois-invariant classes in the $l$-adic cohomology of a variety are explai
ned by algebraic cycles. It is known to imply the conjecture of Birch and
Swinnerton-Dyer (BSD) for elliptic curves over function fields. When the
variety\, now assumed to be in characteristic p\, admits a “partial Fro
benius structure”\, there is a natural extension of Tate’s conjecture.
Assuming this conjecture\, we get not only BSD\, but the following res
ult: the top exterior power of the Mordell-Weil group of an elliptic curv
e is spanned by a “Drinfeld-Heegner” point. This is a report on work
in progress.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Newton
DTSTART:20201007T150000Z
DTEND:20201007T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/13
DESCRIPTION:Title: Evaluating the wild Brauer group\nby Rachel Newton as part of Lon
don number theory seminar\n\n\nAbstract\nThe local-global approach to the
study of rational points on varieties over number fields begins by embeddi
ng the set of rational points on a variety X into the set of its adelic po
ints. The Brauer-Manin pairing cuts out a subset of the adelic points\, ca
lled the Brauer-Manin set\, that contains the rational points. If the set
of adelic points is non-empty but the Brauer-Manin set is empty then we sa
y there's a Brauer-Manin obstruction to the existence of rational points o
n X. Computing the Brauer-Manin pairing involves evaluating elements of th
e Brauer group of X at local points. If an element of the Brauer group has
order coprime to p\, then its evaluation at a p-adic point factors via re
duction of the point modulo p. For p-torsion elements this is no longer th
e case: in order to compute the evaluation map one must know the point to
a higher p-adic precision. Classifying p-torsion Brauer group elements acc
ording to the precision required to evaluate them at p-adic points gives a
filtration which we describe using work of Bloch and Kato. Applications o
f our work include addressing Swinnerton-Dyer's question about which place
s can play a role in the Brauer-Manin obstruction. This is joint work with
Martin Bright.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ziyang Gao
DTSTART:20201014T150000Z
DTEND:20201014T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/14
DESCRIPTION:Title: Bounding the number of rational points on curves\nby Ziyang Gao a
s part of London number theory seminar\n\n\nAbstract\nMazur conjectured\,
after Faltings’s proof of the Mordell conjecture\, that the number of ra
tional points on a curve of genus g at least 2 defined over a number field
of degree d is bounded in terms of g\, d and the Mordell-Weil rank. In pa
rticular the height of the curve is not involved. In this talk I will expl
ain how to prove this conjecture and some generalizations. I will focus on
how functional transcendence and unlikely intersections are applied in th
e proof. If time permits\, I will talk about how the dependence on d can b
e furthermore removed if we moreover assume the relative Bogomolov conject
ure. This is joint work with Vesselin Dimitrov and Philipp Habegger.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rob Kurinczuk (Imperial College London)
DTSTART:20201028T160000Z
DTEND:20201028T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/15
DESCRIPTION:Title: Moduli of Langlands parameters and LLIF\nby Rob Kurinczuk (Imperi
al College London) as part of London number theory seminar\n\n\nAbstract\n
The conjectural local Langlands correspondence connects representations of
p-adic groups to certain representations of Galois groups of local fields
called Langlands parameters. In recent joint work with Dat\, Helm\, and
Moss\, we have constructed moduli spaces of Langlands parameters over Z[1/
p] and studied their geometry. We expect this geometry is reflected in th
e representation theory of the p-adic group. In particular\, our main con
jecture "local Langlands in families" describes the GIT quotient of the mo
duli space of Langlands parameters in terms of the centre of the category
of representations of the p-adic group generalising a theorem of Helm-Moss
for GL(n).\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:João Lourenço
DTSTART:20201111T160000Z
DTEND:20201111T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/16
DESCRIPTION:Title: The Scholze-Weinstein conjecture on local models\nby João Louren
ço as part of London number theory seminar\n\n\nAbstract\nInspired by the
theory of local models of Shimura varieties\, Scholze-Weinstein proposed
a conjecture predicting representability of certain minuscule closed sub-v
-sheaves of their p-adic de Rham affine Grassmannian by a projective flat
and geometrically reduced normal scheme.\n\nIn my talk\, I'll explain the
motivation behind the problem stemming from Shimura varieties\, review the
necessary technical background and ultimately sketch a proof for pseudo-t
ame groups without exceptional factors. To achieve this\, I'll determine t
he Picard group of the Witt vectors affine Grassmannian as conjectured by
Bhatt-Scholze. Time permitting\, I might outline a (very much incomplete)
strategy for handling exceptional groups.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annette Huber (Universität Freiburg)
DTSTART:20201118T160000Z
DTEND:20201118T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/17
DESCRIPTION:Title: Exponential periods and o-minimality\nby Annette Huber (Universit
ät Freiburg) as part of London number theory seminar\n\n\nAbstract\n(join
t work with Johan Commelin and Philipp Habegger)\nRoughly\, period numbers
are defined by integrals of the form\n$\\int_\\sigma\\omega$ with $\\omeg
a$ and $\\sigma$ of algebraic nature.\nThis can be made precise in very di
fferent languages: as values of\nthe period pairing between de Rham cohomo
logy and singular homology\nof algebraic varieties or motives defined over
number fields\, or more\nnaively as\nvolumes of semi-algebraic sets.\n\nM
ore recently\, exponential periods have come into focus. Roughly\, they\na
re of the form $\\int_\\sigma e^{-f}\\omega$ with $\\sigma\,\\omega$ and n
ow\nalso $f$ of algebraic nature. They appear are periods for the Rham com
plex\nof an irregular connection. We want to explain how the "naiv" side o
f\nthe story can be formulated in this case.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Garcia (UCl)
DTSTART:20201209T160000Z
DTEND:20201209T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/20
DESCRIPTION:Title: Eisenstein classes and hyperplane complements\nby Luis Garcia (UC
l) as part of London number theory seminar\n\n\nAbstract\nIn recent years
several authors (Sczech\, Nori\, Hill\, Charollois-Dasgupta-Greenberg\, Be
ilinson-Kings-Levin) have defined and studied certain group cocycles ("Eis
enstein cocycles") in the cohomology of arithmetic groups. I will discuss
how these constructions can be understood in terms of equivariant cohomolo
gy and characteristic classes. This point of view relates the cocycles to
the theta correspondence and leads to generalisations relating the homolog
y of arithmetic groups to algebraic objects such as meromorphic differenti
als on hyperplane complements. I will discuss these generalisations and an
application to critical values of L-functions. \n\nThe talk is based on j
oint work-in-progress with Nicolas Bergeron\, Pierre Charollois and Akshay
Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pedro Lemos (University College London)
DTSTART:20201216T160000Z
DTEND:20201216T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/21
DESCRIPTION:Title: Residual Galois representations of elliptic curves with image in the
normaliser of a non-split Cartan\nby Pedro Lemos (University College L
ondon) as part of London number theory seminar\n\n\nAbstract\nIt is known
that if $p$ is a prime $>37$\, then the image of the residual Galois repre
sentation $\\bar{\\rho}_{E\,p}: G_{\\mathbb{Q}}\\rightarrow {\\rm GL}_2(\\
mathbb{F}_p)$ attached to an elliptic curve $E/\\mathbb{Q}$ without comple
x multiplication is either ${\\rm GL}_2(\\mathbb{F}_p)$\, or is contained
in the normaliser of a non-split Cartan subgroup of ${\\rm GL}_2(\\mathbb{
F}_p)$. I will report on a recent joint work with Samuel Le Fourn where we
improve this result by showing that if $p>1.4\\times 10^7$\, then $\\bar{
\\rho}_{E\,p}$ is either surjective\, or its image is the normaliser of a
non-split Cartan subgroup of ${\\rm GL}_2(\\mathbb{F}_p)$.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lennart Gehrmann (University of Duisburg-Essen / McGill University
)
DTSTART:20201021T150000Z
DTEND:20201021T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/22
DESCRIPTION:Title: L-invariants\, completed cohomology and big principal series\nby
Lennart Gehrmann (University of Duisburg-Essen / McGill University) as par
t of London number theory seminar\n\n\nAbstract\nLet $f$ be a newform of w
eight $2$ that is Steinberg at $p$. Darmon showed that the Fontaine-Mazur
$L$-invariant of the associated local $p$-adic Galois representation can b
e computed in terms of the cohomology of $p$-arithmetic subgroups of the g
roup $PGL_2(\\mathbb{Q})$.\nOn the other hand Breuil showed that one can c
ompute the $f$-isoyptical part of completed cohomology of the modular curv
e in terms of the cohomology of $p$-arithmetic groups.\nIn this talk I wil
l give generalizations of both results to higher rank reductive groups. Th
is is partly joint work with Giovanni Rosso.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sally Gilles
DTSTART:20201104T160000Z
DTEND:20201104T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/23
DESCRIPTION:Title: Period morphisms and syntomic cohomology\nby Sally Gilles as part
of London number theory seminar\n\n\nAbstract\nIn 2017\, Colmez and Nizio
ł proved a comparison theorem between arithmetic p-adic nearby cycles and
syntomic cohomology sheaves. To prove it\, they gave a local construction
using $(\\varphi\,\\Gamma)$-modules theory which allows to reduce the per
iod isomorphism to a comparison theorem between Lie algebras. In this talk
\, I will first give the geometric version of this construction before exp
laining how to globalize it. This period morphism can be used to describe
the étale cohomology of rigid analytic spaces. In particular\, we deduce
the semi-stable conjecture of Fontaine-Jannsen\, which relates the étal
e cohomology of the rigid analytic variety associated to a formal proper s
emi-stable scheme to its Hyodo-Kato cohomology. This result was also prove
d by (among others) Tsuji\, via the Fontaine-Messing map\, and by Česnav
ičius and Koshikawa\, which generalized the proof of the crystalline con
jecture by Bhatt\, Morrow and Scholze. In the second part of the talk\, I
will explain how we can use the previous map to show that the period morph
ism of Tsuji and the one of Česnavičius-Koshikawa are the same.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Graham (Imperial College London)
DTSTART:20210113T160000Z
DTEND:20210113T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/24
DESCRIPTION:Title: Anticyclotomic Euler systems for conjugate self-dual representations
of $GL(2n)$\nby Andrew Graham (Imperial College London) as part of Lon
don number theory seminar\n\n\nAbstract\nAn Euler system is a collection o
f Galois cohomology classes which satisfy certain compatibility relations
under corestriction\, and by constructing an Euler system and relating the
classes to $L$-values\, one can establish instances of the Bloch--Kato co
njecture. In this talk\, I will describe a construction of an anticyclotom
ic Euler system for a certain class of conjugate self-dual automorphic rep
resentations\, which can be seen as a generalisation of the Heegner point
construction. The classes arise from special cycles on unitary Shimura var
ieties and are closely related to the branching law associated with the sp
herical pair $(GL(n) \\times GL(n)\, GL(2n))$. This is joint work with S.W
.A. Shah.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raphael Steiner (ETH Zürich)
DTSTART:20210120T160000Z
DTEND:20210120T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/25
DESCRIPTION:Title: Fourth moments and sup-norms with the aid of theta functions\nby
Raphael Steiner (ETH Zürich) as part of London number theory seminar\n\n\
nAbstract\nIt is a classical problem in harmonic analysis to bound $L^p$-n
orms of eigenfunctions of the Laplacian on (compact) Riemannian manifolds
in terms of the eigenvalue. A general sharp result in that direction was g
iven by Hörmander and Sogge. However\, in an arithmetic setting\, one oug
ht to do better. Indeed\, it is a classical result of Iwaniec and Sarnak t
hat exactly that is true for Hecke-Maass forms on arithmetic hyperbolic su
rfaces. They achieved their result by considering an amplified second mome
nt of Hecke eigenforms. Their technique has since been adapted to numerous
other settings. In my talk\, I shall explain how to use Shimizu's theta f
unction to express a fourth moment of Hecke eigenforms in geometric terms
(second moment of matrix counts). In joint work with Ilya Khayutin and Pau
l Nelson\, we give sharp bounds for said matrix counts and thus a sharp bo
und on the fourth moment in the weight and level aspect. As a consequence\
, we improve upon the best known bounds for the sup-norm in these aspects.
In particular\, we prove a stronger than Weyl-type sub-convexity result.\
n
LOCATION:https://stable.researchseminars.org/talk/LNTS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lynnelle Ye (Stanford University)
DTSTART:20210127T160000Z
DTEND:20210127T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/26
DESCRIPTION:Title: Properness for eigenvarieties\nby Lynnelle Ye (Stanford Universit
y) as part of London number theory seminar\n\n\nAbstract\nCan a family of
finite-slope modular Hecke eigenforms lying over a punctured disc in weigh
t space always be extended over the puncture? This was first asked by Cole
man and Mazur in 1998 and settled by Diao and Liu in 2016 using deep\, pow
erful Galois-theoretic machinery. We will discuss a new proof which is geo
metric and explicit and uses no Galois theory\, and which generalizes in s
ome cases to Hilbert modular forms. We adapt an earlier method of Buzzard
and Calegari based on elementary properties of overconvergent modular form
s\, for which we have to extend the construction of Andreatta-Iovita-Pillo
ni overconvergent forms farther into the supersingular locus.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lassina Dembélé (University of Luxembourg)
DTSTART:20210203T160000Z
DTEND:20210203T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/27
DESCRIPTION:Title: Finite flat $p$-group schemes over $\\mathbf{Z}$\nby Lassina Demb
élé (University of Luxembourg) as part of London number theory seminar\n
\n\nAbstract\nConjecture (Abrashkin-Fontaine): For $p$ prime\, the only si
mple finite flat group schemes of $p$-power order defined over $\\mathbf{Z
}$ are $\\mathbf{Z}/p\\mathbf{Z}$ and $\\mu_p$.\n\nAbrashkin and Fontaine
separately proved that this conjecture is true for $p \\le 17$. In this ta
lk\, we extend their result to the primes $p \\le 37$ under GRH. (This is
joint work with René Schoof.)\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Esteban Rodriguez Camargo (ENS de Lyon)
DTSTART:20210210T160000Z
DTEND:20210210T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/28
DESCRIPTION:Title: Dual Eichler-Shimura maps for the modular curve\nby Juan Esteban
Rodriguez Camargo (ENS de Lyon) as part of London number theory seminar\n\
n\nAbstract\nAndreatta-Iovita-Stevens have constructed interpolations of
the small slope part of the Eichler-Shimura decomposition for the modular
curve. Roughly speaking\, they defined in a geometric way a map from the
overconvergent modular symbols of weight k\, to the overconvergent modular
forms of weight k+2. Then\, using classicality theorems of Coleman and
Ash-Stevens\, they achieved a Hodge-Tate decomposition of the small slope
part of overconvergent modular symbols. On the other hand\, in a recent pa
per of Boxer-Pilloni\, the authors proved that higher Coleman and Hida th
eories exist for the modular curve. The aim of this talk is to construct g
eometrically a map from the higher cohomology of overconvergent modular f
orms of weight -k to the modular symbols as above. We shall recover the Ho
dge-Tate decomposition of the small slope part of modular symbols\, with t
he addition that all the maps involved are defined using the geometry of t
he modular curve. If time permits\, we will discuss the compatibility of t
he previous work with duality.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART:20210217T160000Z
DTEND:20210217T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/29
DESCRIPTION:Title: Beilinson-Bloch conjecture for unitary Shimura varieties\nby Chao
Li (Columbia University) as part of London number theory seminar\n\n\nAbs
tract\nFor certain automorphic representations $\\pi$ on unitary groups\,
we show that if $L(s\, \\pi)$ vanishes to order one at the center $s=1/2$\
, then the associated $\\pi$-localized Chow group of a unitary Shimura var
iety is nontrivial. This proves part of the Beilinson-Bloch conjecture for
unitary Shimura varieties\, which generalizes the BSD conjecture. Assumin
g Kudla's modularity conjecture\, we further prove the arithmetic inner pr
oduct formula for $L'(1/2\, \\pi)$\, which generalizes the Gross-Zagier fo
rmula. We will motivate these conjectures and discuss some aspects of the
proof. We will also mention recent extensions applicable to certain symmet
ric power L-functions of elliptic curves. This is joint work with Yifeng L
iu.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlotte Chan (MIT)
DTSTART:20210224T160000Z
DTEND:20210224T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/30
DESCRIPTION:Title: Geometric L-packets of toral supercuspidal representations\nby Ch
arlotte Chan (MIT) as part of London number theory seminar\n\n\nAbstract\n
In 2001\, Yu gave an algebraic construction of supercuspidal representatio
ns of p-adic groups (now known to be exhaustive when the residual characte
ristic is sufficiently large---Kim\, Fintzen). There has since been a lot
of progress towards explicitly constructing the local Langlands correspond
ence: Kazhdan-Varshavsky and DeBacker-Reeder (depth zero)\, Reeder and DeB
acker-Spice (toral)\, and Kaletha (regular supercuspidals). In this talk\,
we present recent and ongoing work investigating a geometric counterpart
to this story. This is based on joint work with Alexander Ivanov and joint
work with Masao Oi.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anders Södergren (Chalmers University of Technology)
DTSTART:20210303T160000Z
DTEND:20210303T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/31
DESCRIPTION:Title: Cubic fields\, low-lying zeros and the L-functions Ratios Conjecture<
/a>\nby Anders Södergren (Chalmers University of Technology) as part of L
ondon number theory seminar\n\n\nAbstract\nIn this talk I will discuss rec
ent work on the low-lying zeros in the family of $L$-functions attached to
non-Galois cubic Dedekind zeta functions. In particular\, I will describe
the close relation between these low-lying zeros and precise counting res
ults for cubic fields with local conditions. The main application of this
investigation is a conditional omega result for cubic field counting funct
ions. I will also discuss the $L$-functions Ratios Conjecture associated t
o this family of Dedekind zeta functions and the fact that the conjecture
in its standard form does not predict all the terms in the family's one-le
vel density of low-lying zeros. This is joint work with Peter Cho\, Daniel
Fiorilli and Yoonbok Lee.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Min Lee
DTSTART:20210310T160000Z
DTEND:20210310T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/32
DESCRIPTION:Title: Linnik problem for Maass-Hecke cusp forms and effective multiplicity
one theorem\nby Min Lee as part of London number theory seminar\n\n\nA
bstract\nThe strong multiplicity one theorem (for GL(2)\, proved by Jacque
t and Langlands) implies that if two Maass-Hecke cuspforms share the same
Laplacian eigenvalue and the same Hecke eigenvalues for almost all primes
then the two forms must be equal up to a constant multiple. In this talk
we consider the following question\, an analogue of Linnik’s question fo
r Dirichlet characters: if the two forms are not equal up to a constant mu
ltiple\, how large can the first prime p be\, such that the corresponding
Hecke eigenvalues differ? Alternatively we can also ask: how large is the
dimension of the joint eigenspace of the given finite set of Hecke operato
rs and the Laplace operator? We approach these two questions with two diff
erent methods. This is a joint work with Junehyuk Jung.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Lecouturier (Yau Mathematical Sciences Center and Tsinghu
a University (Beijing))
DTSTART:20210317T160000Z
DTEND:20210317T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/33
DESCRIPTION:Title: On an analogue of a conjecture of Sharifi for imaginary quadratic fie
lds\nby Emmanuel Lecouturier (Yau Mathematical Sciences Center and Tsi
nghua University (Beijing)) as part of London number theory seminar\n\n\nA
bstract\nWe explore a relation between the cohomology of certain Bianchi 3
-folds\, modulo some Eisenstein ideal\, to the arithmetic of imaginary qua
dratic fields.\nFor instance\, in the case of Euclidean imaginary quadrati
c fields\, we get a relation between modular symbols and cup-products of e
lliptic units.\nThis is similar to conjectures of Sharifi for classical mo
dular curves\, relating modular symbols to cup-product of cyclotomic units
. \nThis is work in progress with Jun Wang.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshay Venkatesh (Institute for Advanced Study)
DTSTART:20210324T160000Z
DTEND:20210324T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/34
DESCRIPTION:Title: Central L-values up to squares\nby Akshay Venkatesh (Institute fo
r Advanced Study) as part of London number theory seminar\n\n\nAbstract\nT
his is a report on joint work -- in progress -- with A. Abdurrahman. \nGi
ven an everywhere unramified symplectic Galois representation\nover a func
tion field\, we propose a conjectural formula for its central L-value\nup
to squares in the coefficient field\, in terms of a certain cohomological
invariant\nof the representation. \n \nI'll describe three types of evi
dence for this conjecture\, coming\nfrom numerical examples\, topology\, a
nd automorphic forms. \nThen I will discuss (much more speculatively) what
the ramified/number field\nanalogue of the formula might be\, and its pot
ential relationship to a theory\nof "higher epsilon factors."\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Viehmann
DTSTART:20210428T150000Z
DTEND:20210428T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/36
DESCRIPTION:Title: Newton strata on the B$_{dR}$-Grassmannian\nby Eva Viehmann as pa
rt of London number theory seminar\n\n\nAbstract\nRecently\, Fargues and S
cholze laid the foundations for\na geometric Langlands program on the Farg
ues-Fontaine curve. One of the\ncentral objects of interest is the stack B
un$_G$ of $G$-bundles on the\ncurve. I will explain how to determine the u
nderlying topological space\n|Bun$_{G}$| and its relation to the Newton st
ratification on the\nB$_{dR}$-Grassmannian.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitris Koukoulopoulos
DTSTART:20210505T150000Z
DTEND:20210505T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/37
DESCRIPTION:Title: Anatomy of integers\, polynomials and permutations\nby Dimitris K
oukoulopoulos as part of London number theory seminar\n\n\nAbstract\nThere
is a famous analogy between the statistics of the prime factors of a rand
om integer\, of the irreducible factors of a random polynomial over a fini
te field\, and of the cycles of a random permutation. This analogy allows
us to transfer techniques and intuition from one setup to the other\, and
it has been in the center of a lot of recent activity in probabilistic num
ber theory and group theory. I will survey some of this progress\, focusin
g in particular on results about the irreducibility of randomly chosen pol
ynomials with 0\,1 coefficients (joint with Lior Bary-Soroker and Gady Koz
ma)\, as well as on results about the concentration of divisors of random
integers and the size of the Hooley Delta function (joint with Ben Green a
nd Kevin Ford).\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga
DTSTART:20210512T150000Z
DTEND:20210512T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/38
DESCRIPTION:Title: Rational points and Selmer groups of genus 3 curves\nby Jef Laga
as part of London number theory seminar\n\n\nAbstract\nManjul Bhargava and
Arul Shankar have determined the average size of the n-Selmer group of th
e family of all elliptic curves over Q ordered by height\, for n at most 5
. They used this to show that the average rank of elliptic curves is less
than one. \n\nIn this talk we will consider a family of nonhyperelliptic g
enus 3 curves\, and bound the average size of the 2-Selmer group of their
Jacobians. This implies that a majority of curves in this family have rela
tively few rational points. We also consider a family of abelian surfaces
which are not principally polarized and obtain similar results.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Medvedovsky
DTSTART:20210526T150000Z
DTEND:20210526T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/39
DESCRIPTION:Title: Counting modular forms with fixed mod-p Galois representation and Atk
in-Lehner-at-p eigenvalue\nby Anna Medvedovsky as part of London numbe
r theory seminar\n\n\nAbstract\nWork in progress joint with Samuele Anni a
nd Alexandru Ghitza. For N prime to p\, we count the number of classical m
odular forms of level Np and weight k with fixed residual Galois represent
ation and Atkin-Lehner-at-p sign\, generalizing both recent results of Mar
tin generalizing work of Wakatsuki (no residual representation constraint)
and the rhobar-dimension-counting formulas of Bergdall-Pollack and Jochno
witz. To resolve tension between working mod p and the need to invert p\,
we use the trace formula to establish up-to-semisimplifcation isomorphisms
between certain mod-p Hecke\nmodules (namely\, refinements of the weight-
filtration graded pieces W_k) by exhibiting ever-deeper congruences betwee
n traces of prime-power Hecke operators acting on characteristic-zero Heck
e\nmodules. This last technique is new\, purely algebraic\, and may be of
independent interest\; it relies on a combinatorial theorem whose proof be
nefited from a beautiful boost from Gessel.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce
DTSTART:20210602T150000Z
DTEND:20210602T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/40
DESCRIPTION:Title: Counting problems\, from the perspective of moments\nby Lillian P
ierce as part of London number theory seminar\n\n\nAbstract\nMany question
s in number theory can be phrased as counting problems. How many number fi
elds are there? How many elliptic curves are there? How many integral solu
tions to this system of Diophantine equations are there? If the answer is
“infinitely many\,” we want to understand the order of growth for the
number of objects we are counting in the “family." But in many settings
we are also interested in finer-grained questions\, like: how many number
fields are there\, with fixed degree and fixed discriminant? We know the a
nswer is “finitely many\,” but it would have important consequences if
we could show the answer is always “very few indeed.” In this talk\,
we will describe a way that these finer-grained questions can be related t
o the bigger infinite-family questions. Then we will use this perspective
to survey interconnections between several big open conjectures in number
theory\, related in particular to class groups and number fields.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pip Goodman
DTSTART:20210609T150000Z
DTEND:20210609T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/41
DESCRIPTION:Title: Superelliptic curves with large Galois images\nby Pip Goodman as
part of London number theory seminar\n\n\nAbstract\nLet $K$ be a number fi
eld. The inverse Galois problem for $K$ asks if for every finite group $G$
there exists a Galois extension $L/K$ whose Galois group is isomorphic to
$G$. Many people have used torsion points on abelian varieties to realise
symplectic similitude groups (${\\rm GSp}_n(F_\\ell)$) over $Q$.\n\nIn th
is talk\, we examine mod $\\ell$ Galois representations attached to supere
lliptic curves and use them to realise general linear and unitary similitu
de groups over cyclotomic fields. A variety of mathematics is involved\, i
ncluding group theory\, CM theory\, root discriminant bounds\, and models
of curves.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pär Kurlberg
DTSTART:20210616T150000Z
DTEND:20210616T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/42
DESCRIPTION:Title: Level repulsion for arithmetic toral point scatterers\nby Pär Ku
rlberg as part of London number theory seminar\n\n\nAbstract\nThe Seba bil
liard was introduced to study the transition between\n integrability an
d chaos in quantum systems. The model seem to exhibit\n intermediate le
vel statistics with strong repulsion between nearby\n eigenvalues (cons
istent with random matrix theory predictions for\n spectra of chaotic s
ystems)\, whereas large gaps seem to have "Poisson\n tails" (as for spe
ctra of integrable systems.)\n\n We investigate the closely related "to
ral point scatterer"-model\, i.e.\,\n the Laplacian perturbed by a delt
a-potential\, on 3D tori of the form\n R^3/Z^3. This gives a rank one
perturbation of the original Laplacian\,\n and it is natural to split t
he spectrum/eigenspaces into two parts: the\n "old" (unperturbed) one s
panned by eigenfunctions vanishing at the\n scatterer location\, and th
e "new" part (spanned by Green's functions).\n We show that there is st
rong repulsion between the new set of\n eigenvalues.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sug Woo Shin
DTSTART:20210623T100000Z
DTEND:20210623T110000Z
DTSTAMP:20260404T111112Z
UID:LNTS/43
DESCRIPTION:Title: From Langlands–Rapoport to the trace formula\nby Sug Woo Shin a
s part of London number theory seminar\n\n\nAbstract\nIn this talk\, I wil
l report on joint work with Mark Kisin and Yihang Zhu to establish a stabi
lized trace formula computing the cohomology of abelian-type Shimura varie
ties at a prime of good reduction. As a key intermediate step\, we prove a
version of the Langlands-Rapoport conjecture that is more precise than sh
own in Kisin’s recent paper.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Young (Texas A&M University)
DTSTART:20210421T150000Z
DTEND:20210421T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/44
DESCRIPTION:Title: An improved spectral large sieve inequality for $\\text{SL}_3(\\mathb
b Z)$.\nby Matt Young (Texas A&M University) as part of London number
theory seminar\n\n\nAbstract\nI will discuss recent progress on the spectr
al large sieve problem for $\\text{SL}_3(\\mathbb Z)$. The method of proo
f uses duality and its structure reveals unexpected connections to Heath-B
rown's large sieve for cubic characters.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalie Evans
DTSTART:20210630T150000Z
DTEND:20210630T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/45
DESCRIPTION:Title: Correlations of almost primes\nby Natalie Evans as part of London
number theory seminar\n\n\nAbstract\nThe Hardy-Littlewood generalised twi
n prime conjecture states an asymptotic formula for the number of primes $
p\\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While
this conjecture remains wide open\, Matomaki\, Radziwill and Tao proved t
hat it holds on average over $h$\, improving on a previous result of Mikaw
a. In this talk we will discuss an almost prime analogue of the Hardy-Litt
lewood conjecture for which we can go beyond what is known for primes. We
will describe some recent work in which we prove an asymptotic formula for
the number of almost primes $n=p_1p_2 \\le X$ such that $n+h$ has exactly
two prime factors which holds for a very short average over $h$.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vaidehee Thatte (King's College London)
DTSTART:20211208T150000Z
DTEND:20211208T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/48
DESCRIPTION:Title: Understanding the Defect via Ramification Theory\nby Vaidehee Tha
tte (King's College London) as part of London number theory seminar\n\nLec
ture held in Huxley 144\, Imperial.\n\nAbstract\nClassical ramification th
eory deals with complete discrete valuation fields $k((X))$ with perfect r
esidue fields $k$. Invariants such as the Swan conductor capture important
information about extensions of these fields. Many fascinating complicati
ons arise when we allow non-discrete valuations and imperfect residue fiel
ds $k$. Particularly in positive residue characteristic\, we encounter the
mysterious phenomenon of the defect (or ramification deficiency). The occ
urrence of a non-trivial defect is one of the main obstacles to long-stand
ing problems\, such as obtaining resolution of singularities in positive c
haracteristic.\n\nDegree p extensions of valuation fields are building blo
cks of the general case. In this talk\, we will present a generalization o
f ramification invariants for such extensions and discuss how this leads t
o a better understanding of the defect. If time permits\, we will briefly
discuss their connection with some recent work (joint with K. Kato) on upp
er ramification groups.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Tamiozzo (Imperial College London)
DTSTART:20211013T140000Z
DTEND:20211013T150000Z
DTSTAMP:20260404T111112Z
UID:LNTS/50
DESCRIPTION:Title: Perfectoid Jacquet-Langlands and the cohomology of Hilbert modular va
rieties\nby Matteo Tamiozzo (Imperial College London) as part of Londo
n number theory seminar\n\nLecture held in Huxley 139\, Imperial.\n\nAbstr
act\nDeuring and Serre showed that the supersingular locus in a special fi
bre of a modular curve can be identified with a Shimura set attached to a
definite quaternion algebra. I will discuss a perfectoid version of this r
esult over totally real fields\, comparing the cohomology of fibres of the
Hodge-Tate period maps attached to different quaternionic Shimura varieti
es. I will then explain how this can be used to prove vanishing theorems f
or the cohomology with torsion coefficients of Hilbert modular varieties.
This is joint work with Ana Caraiani.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Newton (King's College London)
DTSTART:20211020T140000Z
DTEND:20211020T150000Z
DTSTAMP:20260404T111112Z
UID:LNTS/51
DESCRIPTION:Title: Number fields with prescribed norms\nby Rachel Newton (King's Col
lege London) as part of London number theory seminar\n\nLecture held in Hu
xley 144\, Imperial.\n\nAbstract\nLet $G$ be a finite abelian group\, let
$k$ be a number field\, and let $\\alpha\\in k^\\times$. We count Galois e
xtensions $K/k$ with Galois group $G$ such that $\\alpha$ is a norm from $
K/k$.\nIn particular\, we show that such extensions always exist. This is
joint work with Christopher\nFrei and Daniel Loughran.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Williams (University of Warwick)
DTSTART:20211110T150000Z
DTEND:20211110T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/52
DESCRIPTION:Title: $p$-adic $L$-functions for GL(3)\nby Chris Williams (University o
f Warwick) as part of London number theory seminar\n\nLecture held in Huxl
ey 144\, Imperial.\n\nAbstract\nLet $\\pi$ be a $p$-ordinary cohomological
cuspidal automorphic representation of $GL_n(\\mathbb{A}_\\mathbb{Q})$. A
conjecture of Coates--Perrin-Riou predicts that the (twisted) critical va
lues of its $L$-function $L(\\pi \\times \\chi\,s)$\, for Dirichlet charac
ters $\\chi$ of $p$-power conductor\, satisfy systematic congruence proper
ties modulo powers of $p$\, captured in the existence of a $p$-adic $L$-fu
nction. For $n = 1\,2$ this conjecture has been known for decades\, but fo
r $n \\geq 3$ it is known only in special cases\, e.g. symmetric squares o
f modular forms\; and in all known cases\, $\\pi$ is a functorial transfer
from a proper subgroup of $GL_n$. I will explain what a $p$-adic $L$-func
tion is\, state the conjecture more precisely\, and then report on ongoing
joint work with David Loeffler\, in which we prove this conjecture for $n
=3$ (without any transfer or self-duality assumptions).\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Torzewski (King's College London)
DTSTART:20211124T150000Z
DTEND:20211124T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/53
DESCRIPTION:Title: Lawrence-Venkatesh in families\nby Alex Torzewski (King's College
London) as part of London number theory seminar\n\nLecture held in Huxley
144\, Imperial.\n\nAbstract\nWe outline how the method of Lawrence-Venkat
esh can be used in families to obtain upper bounds on the number of ration
al points on curves of genus > 1 depending only on the reduction modulo a
well chosen prime and the primes of bad reduction. This was first shown by
Faltings as a consequence of the Mordell and Shafarevich Conjectures.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Sempliner (Imperial College London)
DTSTART:20211201T150000Z
DTEND:20211201T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/54
DESCRIPTION:Title: On the almost-product structure on the moduli of bounded global $G$-s
htuka\nby Jack Sempliner (Imperial College London) as part of London n
umber theory seminar\n\nLecture held in Huxley 144\, Imperial.\n\nAbstract
\nLet $X$ be an algebraic curve over $\\mathbb{F}_q$ and $G$ be a reductiv
e algebraic group over $\\mathbb{F}_q(X)$. Under mild technical hypotheses
we construct families of stacks over the moduli $\\text{Sht}_{G\, X\, I}^
{\\mu_*}$ of bounded global $G$-shtuka (a small generalization of the stac
ks studied by Lafforgue and Varshavsky) which provide natural analogues of
Igusa varieties in the function field setting. Our main result is an isom
orphism between certain Igusa varieties associated to moduli of shtuka for
reductive groups $G\, G'$ which are related by an inner twist. Along the
way we prove an almost-product formula computing the compactly supported c
ohomology of the special fibers of $\\text{Sht}_{G\, X\, I}^{\\mu_*}$ with
trivial coefficients in terms of the cohomology of our Igusa stacks and a
function-field analogue of Rapoport-Zink spaces constructed in previous w
ork of Hartl and Arasteh Rad.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rosa Winter
DTSTART:20211117T150000Z
DTEND:20211117T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/55
DESCRIPTION:Title: Density of rational points on del Pezzo surfaces of degree 1\nby
Rosa Winter as part of London number theory seminar\n\nLecture held in Hux
ley 144\, Imperial.\n\nAbstract\nDel Pezzo surfaces are surfaces classifie
d by their degree $d$\, which is an integer between 1 and 9 (for $d\\geq3$
\, these are the smooth surfaces of degree $d$ in $\\mathbb{P}^d$). For de
l Pezzo surfaces of degree at least 2 over a field $k$\, we know that the
set of $k$-rational points is Zariski dense provided that the surface has
one $k$-rational point to start with (that lies outside a specific subset
of the surface for degree 2). However\, for del Pezzo surfaces of degree 1
over a field $k$\, even though we know that they always contain at least
one $k$-rational point\, we do not know if the set of $k$-rational points
is Zariski dense in general. I will talk about a result that is joint work
with Julie Desjardins\, in which we give sufficient conditions for the se
t of $k$-rational points on a specific family of del Pezzo surfaces of deg
ree 1 to be Zariski dense\, where $k$ is any infinite field of characteris
tic 0. These conditions are necessary if $k$ is finitely generated over $\
\mathbb{Q}$. I will compare this to previous results.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Zerbes (University College London)
DTSTART:20211103T150000Z
DTEND:20211103T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/56
DESCRIPTION:Title: Euler systems and the BSD conjecture for abelian surfaces\nby Sar
ah Zerbes (University College London) as part of London number theory semi
nar\n\nLecture held in Huxley 144\, Imperial.\n\nAbstract\nEuler systems a
re one of the most powerful tools for proving cases of the Bloch--Kato con
jecture\, and other related problems such as the Birch and Swinnerton-Dyer
conjecture. I will recall a series of recent works (variously joint with
Loeffler\, Pilloni\, Skinner) giving rise to an Euler system in the cohomo
logy of Shimura varieties for GSp(4)\, and an explicit reciprocity law rel
ating this to values of L-functions. I will then explain work in progress
with Loeffler\, in which we use this Euler system to prove new cases of th
e BSD conjecture for modular abelian surfaces over Q\, and modular ellipti
c curves over imaginary quadratic fields.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Jossen
DTSTART:20220112T160000Z
DTEND:20220112T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/57
DESCRIPTION:Title: A non-hypergeometric E-function\nby Peter Jossen as part of Londo
n number theory seminar\n\n\nAbstract\nWith the goal of generalising the t
heorems of Hermite\, Lindemann\, and Weierstrass about transcendence of va
lues of the exponential function\, Siegel\nintroduced the notion of E-func
tion in his landmark 1929 paper "Über einige Anwendungen diophantischer A
pproximationen". Hypergeometric functions\nprovide a rich class of E-funct
ions\, and Siegel asked whether in fact every E-function is a polynomial e
xpression in hypergeometric E-functions. In a\njoint work with Javier Fres
án\, we answer Siegel's question in the negative.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Yafaev
DTSTART:20220119T160000Z
DTEND:20220119T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/58
DESCRIPTION:Title: Heights of special points and the Andre-Oort conjecture\nby Andre
i Yafaev as part of London number theory seminar\n\n\nAbstract\nThe Andre-
Oort conjecture states that components of the Zariski closure of a set of\
nspecial points in a Shimura variety\, are special subvarieties.\nThis con
jecture has been a subject of active research in \nrecent years.\nThe last
remaining step was to obtain lower bounds for Galois\ndegrees of special
points.\n\nIn a joint work with Gal Biniyamini and Harry Schmidt\, we have
formulated a conjecture \non heights of special points and deduced from i
t the required bounds.\nVery recently\, J.Pila\, A.Shankar and J.Tsimerman
\nannounced a proof of our height conjecture\, thus completing the proof\
nof the Andre-Oort conjecture in full generality.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Caleb Springer
DTSTART:20220126T160000Z
DTEND:20220126T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/59
DESCRIPTION:Title: Every finite abelian group arises as the group of rational points of
an ordinary abelian variety over $\\mathbb{F}_2$\, $\\mathbb{F}_3$\, and
$\\mathbb{F}_5$\nby Caleb Springer as part of London number theory sem
inar\n\n\nAbstract\nWe will show that every finite abelian group arises as
the group of rational points of an ordinary abelian variety over a finite
field with 2\, 3 or 5 elements. Similar results hold over finite fields
of larger cardinality. On our way to proving these results\, we will view
the group of rational points of an abelian variety as a module over its e
ndomorphism ring. By describing this module structure in important cases\,
we obtain (a fortiori) an understanding of the underlying groups. Combini
ng this description of structure with recent results on the cardinalities
of groups of rational points of abelian varieties over finite fields\, we
will deduce the main theorem. This work is joint with Stefano Marseglia.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Walker
DTSTART:20220202T160000Z
DTEND:20220202T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/60
DESCRIPTION:Title: The Average Number of Divisors of the Output of a Quadratic Polynomia
l\nby Alex Walker as part of London number theory seminar\n\n\nAbstrac
t\nLet $d(n)$ count the number of divisors of $n$. In 1963\, Hooley studie
d partial sums $n < X$ of $d(n^2+h)$ and showed that the result was asympt
otic to $c X \\log X + c' X + O(X^{8/9})$ as $X$ tends to infinity (assumi
ng $h$ not a negative square). In other words\, the irreducible polynomial
$Q(x) = x^2 + h$ has outputs with\, on average\, $\\sim \\log x$ many div
isors. Hooley's error bound was improved by Bykoskii in 1987 to $O(X^{2/3}
)$ using the spectral theory of automorphic forms. This talk describes a n
ew proof of Bykovskii's result in a new framework\, now using Dirichlet se
ries and automorphic forms of half-integral weight. This new framework has
limitations but is also quite flexible. To demonstrate this\, we develop
in tandem counts for the average number of divisors of $Q(x\,y) = x^2+y^2+
h$ for $x^2+y^2+h < X$.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Streeter
DTSTART:20220209T160000Z
DTEND:20220209T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/61
DESCRIPTION:Title: Weak approximation for del Pezzo surfaces of low degree\nby Sam S
treeter as part of London number theory seminar\n\n\nAbstract\nConjectural
ly\, the rational points of a del Pezzo surface over a number field are we
ll-distributed among the local points over all but finitely completions of
the ground field—that is\, the surface satisfies weak weak approximatio
n. However\, describing the rational points becomes harder as the degree o
f the del Pezzo surface decreases. As such\, many questions remain unanswe
red for del Pezzo surfaces of low degree. In this talk\, I will report on
recent joint work with Julian Demeio\, in which we prove that del Pezzo su
rfaces of degrees 1 and 2 satisfy weak weak approximation\, provided that
we assume some additional geometric structure in the form of conic fibrati
ons.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nirvana Coppola (Vrije Universiteit Amsterdam)
DTSTART:20220223T160000Z
DTEND:20220223T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/62
DESCRIPTION:Title: Coleman integrals over number fields: a computational approach\nb
y Nirvana Coppola (Vrije Universiteit Amsterdam) as part of London number
theory seminar\n\n\nAbstract\nOne of the deepest mathematical results is F
altings's Theorem on the finiteness of rational points on an algebraic cur
ve of genus $g \\geq 2$. A much more difficult question\, still not comple
tely answered\, is whether given a curve of genus $g \\geq 2$\, we can fin
d all its rational points\, or\, more in general\, all points defined over
a certain number field. An entire (currently very active!) area of resear
ch is devoted to find an answer to such questions\, using the "method of C
habauty".\n\nIn this seminar\, I will talk about one of the first tools em
ployed in Chabauty method\, namely Coleman integrals\, which Coleman used
to compute an explicit bound on the number of rational points on a curve.
After explaining how this is defined\, I will give a generalisation of thi
s definition for curves defined over number fields\, and explain how to ex
plicitly compute these integrals. This is based on an ongoing project\, wh
ich started during the Arizona Winter School 2020\, joint with E. Kaya\, T
. Keller\, N. Müller\, S. Muselli.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Miriam Norris (King's College London)
DTSTART:20220302T160000Z
DTEND:20220302T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/63
DESCRIPTION:Title: Lattice graphs for representations of $GL_3(\\F_p)$\nby Miriam No
rris (King's College London) as part of London number theory seminar\n\n\n
Abstract\nIn a recent paper Le\, Le Hung\, Levin and Morra proved a genera
lisation of Breuil's Lattice conjecture in dimension three. This involved
showing that lattices inside representations of $GL_3(\\F_p)$ coming from
both a global and a local construction coincide. Motivated by this we cons
ider the following graph. For an irreducible representation $\\tau$ of a g
roup $G$ over a finite extension $K$ of $\\Q_p$ we define a graph on the $
\\mathcal{O}_K$-lattices inside $\\tau$ whose edges encapsulate the relati
onship between lattices in terms of irreducible modular representations of
$G$ (or Serre weights in the context of the paper by Le et al.). \n\nIn t
his talk\, I will demonstrate how one can apply the theory of graduated or
ders and their lattices\, established by Zassenhaus and Plesken\, to under
stand the lattice graphs of residually multiplicity free representation ov
er suitably large fields in terms of a matrix called an exponent matrix. F
urthermore I will explain how I have been able to show that one can determ
ine the exponent matrices for suitably generic representation go $GL_3(\\F
_p)$ allowing us to construct their lattice graphs.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:María Inés de Frutos Fernández (Imperial College London)
DTSTART:20220309T160000Z
DTEND:20220309T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/64
DESCRIPTION:Title: Formalizing the ring of adèles and some applications in Lean\nby
María Inés de Frutos Fernández (Imperial College London) as part of Lo
ndon number theory seminar\n\n\nAbstract\nI will present a formalization o
f the ring of adèles and group of idèles of a global field in the Lean 3
theorem prover. Lean is an interactive theorem prover with an ever-growin
g mathematics library. I will give a quick introduction to Lean and explai
n how these definitions were formalized\, with a focus on the kind of deci
sions one has to make during the formalization process.\n\nBesides the def
inition of the adèles\, we will discuss the formalization of applications
including the statement of the main theorem of global class field theory
and a proof that the ideal class group of a number field is isomorphic to
an explicit quotient of its idèle class group.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Rockwood (University of Warwick)
DTSTART:20220316T160000Z
DTEND:20220316T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/65
DESCRIPTION:Title: Spherical varieties and non-ordinary families of cohomology classes\nby Robert Rockwood (University of Warwick) as part of London number th
eory seminar\n\n\nAbstract\nThe theory of norm compatible cohomology class
es is of fundamental importance in Iwasawa theory\, encompassing both the
theory of Euler systems and p-adic L-functions. Loeffler has developed a s
ystematic approach to constructing norm-compatible classes using the theor
y of spherical varieties. We show that classes constructed in this way var
y naturally in Coleman families and give some concrete applications.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Magee
DTSTART:20220323T160000Z
DTEND:20220323T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/66
DESCRIPTION:Title: The maximal spectral gap of a hyperbolic surface\nby Michael Mage
e as part of London number theory seminar\n\n\nAbstract\nA hyperbolic surf
ace is a surface with metric of constant curvature -1. The spectral gap be
tween\nthe first two eigenvalues of the Laplacian on a closed hyperbolic s
urface contains a good deal of\ninformation about the surface\, including
its connectivity\, dynamical properties of its geodesic flow\,\nand error
terms in geodesic counting problems. For arithmetic hyperbolic surfaces th
e spectral gap\nis also the subject of one of the biggest open problems in
automorphic forms: Selberg’s eigenvalue\nconjecture.\nIt was an open pr
oblem from the 1970s whether there exist a sequence of closed hyperbolic s
ur-\nfaces with genera tending to infinity and spectral gap tending to 1/4
. (The value 1/4 here is the\nasymptotically optimal one.) Recently we pro
ved that this is indeed possible. I’ll discuss the very\ninteresting bac
kground of this problem in detail as well as some ideas of the proof. This
is joint work\nwith Will Hide.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Best (VU Amsterdam)
DTSTART:20220427T150000Z
DTEND:20220427T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/67
DESCRIPTION:Title: The S-unit equation and non-abelian Chabauty in depth 2\nby Alex
Best (VU Amsterdam) as part of London number theory seminar\n\nLecture hel
d in Bush House S2.03\, King's College London.\n\nAbstract\nThe S-unit equ
ation is a classical and well-studied Diophantine equation\, with numerous
connections to other Diophantine problems.\nRecent work of Kim and refine
ments due to Betts-Dogra have suggested new cohomological strategies to fi
nd rational and integral points on curves\, based on but massively extendi
ng the classical method of Chabauty. At present\, these methods are only c
onjecturally guaranteed to succeed in general\, but they promise several a
pplications in arithmetic geometry if they could be proved to always work.
\nIn order to better understand the conjectures of Kim that suggest that t
his method should work\, we consider the case of the thrice punctured proj
ective line\, in "depth 2"\, the "smallest" non-trivial extension of the c
lassical method. In doing so we get very explicit results for some S-unit
equations\, demonstrating the usability of the aforementioned cohomologica
l methods in this setting. To do this we determine explicitly equations fo
r (maps between) the (refined) Selmer schemes defined by Kim\, and Betts-D
ogra\, which turn out to have some particularly simple forms.\nThis is joi
nt work with Alexander Betts\, Theresa Kumpitsch\, Martin Lüdtke\, Angus
McAndrew\, Lie Qian\, Elie Studnia\, and Yujie Xu .\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aled Walker (King's College London)
DTSTART:20220504T150000Z
DTEND:20220504T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/68
DESCRIPTION:Title: Correlations of sieve weights and distributions of zeros\nby Aled
Walker (King's College London) as part of London number theory seminar\n\
nLecture held in King's Building K0.18\, King's College London.\n\nAbstrac
t\nIn this talk we will discuss Montgomery's pair correlation conjecture f
or the zeros of the Riemann zeta function. This is a deep spectral conject
ure\, closely related to several arithmetic conjectures on the distributio
n of primes. For example\, even assuming a strong form of the twin prime c
onjecture\, one would only resolve Montgomery's conjecture in a limited ra
nge. Yet\, building on work of Goldston and Gonek from the late 1990s\, we
will present a recent conditional lower bound on the Fourier transform of
Montgomery's pair correlation function\, valid under milder hypotheses. T
he new technical ingredient is a correlation estimate for the Selberg siev
e weights\, for which the level of support of the weights lies beyond the
classical square-root barrier.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Zenz (McGill)
DTSTART:20220511T150000Z
DTEND:20220511T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/69
DESCRIPTION:Title: Holomorphic Hecke Cusp Forms and Quantum Chaos\nby Peter Zenz (Mc
Gill) as part of London number theory seminar\n\nLecture held in King's Bu
ilding\, K0.18.\n\nAbstract\nArithmetic Quantum Chaos (AQC) is an active a
rea of research at the intersection of number theory and physics. One majo
r goal in AQC is to study the mass distribution and behaviour of Hecke Maa
ss cusp forms on hyperbolic surfaces as the Laplace eigenvalue tends to in
finity. In this talk we will focus on analogous questions for holomorphic
Hecke cusp forms. First\, we will review some of the important solved and
unsolved questions in the area\, like the Quantum Unique Ergodicity proble
m or the Gaussian Moment Conjecture. We then elaborate on a sharp bound fo
r the fourth moment of holomorphic cusp forms and ongoing work on evaluati
ng the averaged sixth moment of holomorphic cusp forms. These are special
instances of the Gaussian Moment Conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emilia Alvarez (Bristol)
DTSTART:20220518T150000Z
DTEND:20220518T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/70
DESCRIPTION:Title: Moment computations in the classical compact ensembles\nby Emilia
Alvarez (Bristol) as part of London number theory seminar\n\n\nAbstract\n
After a brief introduction on the random matrix applications to number the
ory\, I will present a collection of moment computations over the unitary\
, symplectic and special orthogonal random matrix ensembles that I've done
throughout my thesis. I will highlight work on the asymptotics of moments
of the logarithmic derivative of characteristic polynomials evaluated nea
r the point 1. Throughout\, the focus will be on the methods used\, the mo
tivation from number theory and directions for future work.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Justin Trias-Batle (Imperial College London)
DTSTART:20220525T150000Z
DTEND:20220525T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/71
DESCRIPTION:Title: Towards a theta correspondence in families for type II dual pairs
\nby Justin Trias-Batle (Imperial College London) as part of London number
theory seminar\n\nLecture held in K0.18 King's building KCL.\n\nAbstract\
nThis is current work with Gil Moss. The classical local theta corresponde
nce for p-adic reductive dual pairs defines a bijection between prescribed
subsets of irreducible smooth complex representations coming from two gro
ups (H\,H')\, forming a dual pair in a symplectic group. Alberto Mínguez
extended this result for type II dual pairs\, i.e. when (H\,H') is made of
general linear groups\, to representations with coefficients in an algebr
aically closed field of characteristic l as long as the characteristic l d
oes not divide the pro-orders of H and H'. For coefficients rings like Z[1
/p]\, we explain how to build a theory in families for type II dual pairs
that is compatible with reduction to residue fields of the base coefficien
t ring\, where central to this approach is the integral Bernstein centre.
We translate some weaker properties of the classical correspondence\, such
as compatibility with supercuspidal support\, as a morphism between the i
ntegral Bernstein centres of H and H' and interpret it for the Weil repres
entation. In general\, we only know that this morphism is finite though we
may expect it to be surjective. This would result in a closed immersion b
etween the associated affine schemes as well as a correspondence between c
haracters of the Bernstein centre.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raman Parimala (Emory)
DTSTART:20220601T150000Z
DTEND:20220601T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/72
DESCRIPTION:Title: The Brauer group of hyperelliptic curves over number fields\nby R
aman Parimala (Emory) as part of London number theory seminar\n\nLecture h
eld in King's building K0.18\, King's College London.\n\nAbstract\nWe disc
uss period-index bounds for the unramified Brauer group of function fields
of hyperelliptic curves over number fields. We describe a link to the qu
estion of Hasse principle for smooth intersection of quadrics.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sacha Mangerel (Durham)
DTSTART:20220615T150000Z
DTEND:20220615T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/73
DESCRIPTION:Title: Gaussian distribution of squarefree and B-free numbers in short inter
vals\nby Sacha Mangerel (Durham) as part of London number theory semin
ar\n\nLecture held in Room K0.18 in the King's Building.\n\nAbstract\n(Joi
nt with O. Gorodetsky and B. Rodgers) It is of classical interest in analy
tic number theory to understand the fine-scale distribution of arithmetic
sequences such as the primes. For a given length scale h\, the number of e
lements of a ``nice'' sequence in a uniformly randomly selected interval $
(x\,x+h]\, 1 \\leq x \\leq X$\, might be expected to follow the statistics
of a normally distributed random variable (in suitable ranges of $1 \\leq
h \\leq X$). Following the work of Montgomery and Soundararajan\, this i
s known to be true for the primes\, but only if we assume several deep and
long-standing conjectures among which the Riemann Hypothesis. \n\nAs a mo
del for the primes\, in this talk I will address such statistical question
s for the sequence of squarefree numbers\, i.e.\, numbers not divisible by
the square of any prime\, among other related ``sifted'' sequences called
B-free numbers. I hope to further motivate and explain our main result th
at shows\, unconditionally\, that short interval counts of squarefree numb
ers do satisfy Gaussian statistics\, answering several questions of R.R. H
all.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lassina Dembele (King's College London)
DTSTART:20220622T150000Z
DTEND:20220622T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/74
DESCRIPTION:Title: Explicit inertial local Langlands correspondence for ${\\rm GL_2}$ an
d arithmetic applications\nby Lassina Dembele (King's College London)
as part of London number theory seminar\n\n\nAbstract\nIn this talk\, we d
escribe an algorithm for computing automorphic and inertial types for ${\\
rm GL_2}$\, and gives several applications. (This is joint work with Nuno
Freitas and John Voight.)\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yukako Kezuka (Jussieu)
DTSTART:20220629T150000Z
DTEND:20220629T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/75
DESCRIPTION:Title: Arithmetic of elliptic curves with complex multiplication at small pr
imes\nby Yukako Kezuka (Jussieu) as part of London number theory semin
ar\n\nLecture held in King's Building K0.18.\n\nAbstract\nThe equation E:
x^3+y^3=N defines a classical family of elliptic curves as N varies over c
ube-free positive integers. They admit complex multiplication\, which allo
ws us to tackle the conjecture of Birch and Swinnerton-Dyer for E effectiv
ely. Indeed\, using Iwasawa theory\, Rubin was able to show the p-part of
the conjecture for E for all primes p\, except for the primes 2 and 3. The
theory becomes much more complex at these small primes\, but at the same
time we can observe some interesting phenomena. I will explain a method to
study the p-adic valuation of the algebraic part of the central L-value o
f E\, and I will establish the 3-part of the conjecture for E in special c
ases. I will then explain a relation between the 2-part of a certain ideal
class group and the Tate-Shafarevich group of E. Part of this talk is bas
ed on joint work with Yongxiong Li.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Si Ying Lee (MPIM Bonn)
DTSTART:20221012T150000Z
DTEND:20221012T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/76
DESCRIPTION:Title: Eichler-Shimura relations of Hodge type Shimura varieties\nby Si
Ying Lee (MPIM Bonn) as part of London number theory seminar\n\nLecture he
ld in Lecture held in Huxley 144\, Imperial.\n\nAbstract\nThe well-known c
lassical Eichler-Shimura relation for modular curves asserts that the Heck
e operator $T_p$ is equal\, as an algebraic correspondence over the specia
l fiber\, to the sum of Frobenius and Verschiebung. Blasius and Rogawski p
roposed a generalization of this result for Shimura varieties with good re
duction at $p$\, and conjectured that the Frobenius satisfies a certain He
cke polynomial. I will talk about a recent proof of this for some Shimura
varieties of Hodge type.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peiyi Cui (University of East Anglia)
DTSTART:20221102T160000Z
DTEND:20221102T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/77
DESCRIPTION:Title: A decomposition of the category of l-modular representations of SL_n(
F).\nby Peiyi Cui (University of East Anglia) as part of London number
theory seminar\n\nLecture held in Huxley 139\, Imperial.\n\nAbstract\nLet
F be a p-adic field\, and k an algebraically closed field of characterist
ic l different from p. In this talk\, we will first give a category decomp
osition of Rep_k(SL_n(F))\, the category of smooth k-representations of SL
_n(F)\, with respect to the GL_n(F)-equivalent supercuspidal classes of SL
_n(F)\, which is not always a block decomposition in general. We then give
a block decomposition of the supercuspidal subcategory\, by introducing a
partition on each GL_n(F)-equivalent supercuspidal class through type the
ory\, and we interpret this partition by the sense of l-blocks of finite g
roups. We give an example where a block of Rep_k(SL_2(F)) is defined with
respect to several SL_2(F)-equivalent supercuspidal classes\, which is dif
ferent from the case where l is zero. We end this talk by giving a predict
ion on the block decomposition of Rep_k(A) for a general p-adic group A.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu Min (Imperial College London)
DTSTART:20221116T160000Z
DTEND:20221116T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/78
DESCRIPTION:Title: Hodge--Tate crystals and Sen theory\nby Yu Min (Imperial College
London) as part of London number theory seminar\n\nLecture held in Huxley
139\, Imperial.\n\nAbstract\nLet $K$ be a finite extension of $\\mathbb Q_
p$. Bhatt and Scholze have proved that the category of prismatic $F$-cryst
als on the absolute prismatic site of $\\mathcal O_K$ is equivalent to the
category of crystalline $\\mathbb Z_p$-representations of the absolute Ga
lois group of $K$. In this talk\, we will instead consider the (rational)
Hodge--Tate crystals on the absolute prismatic site of $\\mathcal O_K$ or
more generally of a smooth $p$-adic formal scheme. We will show how Hodge-
-Tate crystals are related to the Sen theory. If time permits\, we will al
so discuss its application in the arithmetic $p$-adic Simpson corresponden
ce. This is joint work with Yupeng Wang.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Manning (Imperial College London)
DTSTART:20221123T160000Z
DTEND:20221123T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/79
DESCRIPTION:Title: The Wiles-Lenstra-Diamond numerical criterion over imaginary quadrati
c fields\nby Jeff Manning (Imperial College London) as part of London
number theory seminar\n\nLecture held in Huxley 139\, Imperial.\n\nAbstrac
t\nWiles' modularity lifting theorem was the central argument in his proof
of modularity of (semistable) elliptic curves over Q\, and hence of Ferma
t's Last Theorem. His proof relied on two key components: his "patching" a
rgument (developed in collaboration with Taylor) and his numerical isomorp
hism criterion.\n\nIn the time since Wiles' proof\, the patching argument
has been generalized extensively to prove a wide variety of modularity lif
ting results. In particular Calegari and Geraghty have found a way to gene
ralize it to prove potential modularity of elliptic curves over imaginary
quadratic fields (contingent on some standard conjectures). The numerical
criterion on the other hand has proved far more difficult to generalize\,
although in situations where it can be used it can prove stronger results
than what can be proven purely via patching.\n\nIn this talk I will presen
t joint work with Srikanth Iyengar and Chandrashekhar Khare which proves a
generalization of the numerical criterion to the context considered by Ca
legari and Geraghty (and contingent on the same conjectures). This allows
us to prove integral "R=T" theorems at non-minimal levels over imaginary q
uadratic fields\, which are inaccessible by Calegari and Geraghty's method
. The results provide new evidence in favor of a torsion analog of the cla
ssical Langlands correspondence.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Toby Gee (IC)
DTSTART:20221130T160000Z
DTEND:20221130T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/80
DESCRIPTION:Title: Congruences between modular forms and the categorical p-adic Langland
s program\nby Toby Gee (IC) as part of London number theory seminar\n\
nLecture held in Huxley 139\, Imperial.\n\nAbstract\nI will attempt to giv
e a gentle introduction to the categorical p-adic Langlands program and it
s connections to questions about congruences between modular forms.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hanneke Wiersema (University of Cambridge)
DTSTART:20221207T160000Z
DTEND:20221207T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/81
DESCRIPTION:Title: Modularity in the partial weight one case\nby Hanneke Wiersema (U
niversity of Cambridge) as part of London number theory seminar\n\nLecture
held in Huxley 139\, Imperial.\n\nAbstract\nThe strong form of Serre's co
njecture states that a two-dimensional mod $p$ representation of the absol
ute Galois group of $\\mathbb{Q}$ arises from a modular form of a specific
weight\, level and character. Serre restricted to modular forms of weight
at least 2\, but Edixhoven later refined this conjecture to include weigh
t one modular forms. In this talk we explore analogues of Edixhoven's refi
nement for Galois representations of totally real fields\, extending recen
t work of Diamond–-Sasaki. In particular\, we show how modularity of par
tial weight one Hilbert modular forms can be related to modularity of Hilb
ert modular forms with regular weights\, and vice versa. Time permitting\,
we will also discuss a $p$-adic Hodge theoretic version of this.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rong Zhou (University of Cambridge)
DTSTART:20221214T160000Z
DTEND:20221214T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/82
DESCRIPTION:Title: Independence of $\\ell$ for $G$-valued Weil--Deligne representations
associated to abelian varieties\nby Rong Zhou (University of Cambridge
) as part of London number theory seminar\n\nLecture held in Huxley 139\,
Imperial.\n\nAbstract\nLet $A$ be an abelian variety over a number field $
E$ of dimension $g$ and $\\rho_\\ell:\\mathrm{Gal}(\\overline{E}/E)\\right
arrow \\mathrm{GL}_{2g}(\\mathbb{Q}_\\ell)$ the Galois representation on t
he $\\ell$-adic Tate module of $A$. For a place $v$ of $E$ not dividing $\
\ell$\, upon fixing an isomorphism $\\overline{\\mathbb{Q}}_\\ell\\cong \\
mathbb{C}$\, Grothendieck’s $\\ell$-adic monodromy theorem associates to
$\\rho_\\ell$ a $\\mathrm{GL}_{2g}(\\mathbb{C})$-valued Weil-Deligne repr
esentation $\\rho_{\\ell\,v}^{WD}$. Then it is known that the conjugacy cl
ass of $\\rho_{\\ell\,v}^{WD}$ is defined over $\\mathbb{Q}$ and independe
nt of $\\ell.$ When $v$ is a place a good reduction\, this is just the res
ult that the characteristic polynomial of Frobenius is defined over $\\mat
hbb{Z}$ and independent of $\\ell$.\n\nWe consider a refinement of this re
sult. A Theorem of Deligne implies that upon replacing $E$ by a finite ext
ension\, the representations $\\rho_{\\ell\,v}^{WD}$ can be refined to a $
G(\\mathbb{C})$-valued Weil-Deligne representation $\\rho^{WD\,G}_{\\ell\,
v}$\, where $G$ is the Mumford--Tate group of $A$. We prove that for $p>2$
and $v|p$ a place of $E$ where $A$ has semistable reduction\, the conjuga
cy class of $\\rho^{WD\,G}_{\\ell\,v}$ is defined over $\\mathbb{Q}$ and i
ndependent of $\\ell$. This is joint work with Mark Kisin.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (University of Cambridge)
DTSTART:20221026T150000Z
DTEND:20221026T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/83
DESCRIPTION:Title: Symmetric power functoriality for Hilbert modular forms\nby Jack
Thorne (University of Cambridge) as part of London number theory seminar\n
\nLecture held in Huxley 139\, Imperial.\n\nAbstract\nSymmetric power func
toriality is one of the basic cases of Langlands' functoriality conjecture
s and is the route to the proof of the Sato-Tate conjecture (concerning th
e distribution of the modulo $p$ point counts of an elliptic curve over $\
\mathbb{Q}$\, as the prime $p$ varies).\n\nI will discuss the proof of the
existence of the symmetric power liftings of Hilbert modular forms of reg
ular weight. This is joint work with James Newton.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chung-Hang (Kevin) Kwan (University College London (UCL))
DTSTART:20221019T150000Z
DTEND:20221019T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/84
DESCRIPTION:Title: Moments and Periods for GL(3)\nby Chung-Hang (Kevin) Kwan (Univer
sity College London (UCL)) as part of London number theory seminar\n\nLect
ure held in Huxley 144\, Imperial.\n\nAbstract\nIn the past century\, mome
nts of L-functions have been important in number theory and are well-motiv
ated by a variety of arithmetic applications. In this talk\, we will begin
with two elementary counting problems of Diophantine nature as motivation
\, followed by a survey of techniques in the past and the present. The mai
n goal is to demonstrate how period integrals can be used to study moments
of automorphic L-functions and uncover the interesting underlying structu
res (some of them can be modeled by random matrix theory).\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Boxer (Imperial College London)
DTSTART:20221109T160000Z
DTEND:20221109T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/85
DESCRIPTION:Title: Higher Hida theory for Siegel modular varieties\nby George Boxer
(Imperial College London) as part of London number theory seminar\n\nLectu
re held in Huxley 139\, Imperial.\n\nAbstract\nThe goal of higher Hida the
ory is to study the ordinary part of coherent cohomology of Shimura variet
ies integrally. We introduce a higher coherent cohomological analog of Hi
da's space of ordinary p-adic modular forms\, which is defined as the "ord
inary part" of the coherent cohomology with "partial compact support" of t
he ordinary Igusa variety. Then we give an analog of Hida's classicality
theorem in this setting. This is joint work with Vincent Pilloni.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Green (University College London)
DTSTART:20230111T160000Z
DTEND:20230111T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/86
DESCRIPTION:Title: An arithmetic analogue of the parity conjecture\nby Holly Green (
University College London) as part of London number theory seminar\n\nLect
ure held in Rm. 505\, UCL Department of Mathematics (UCL Union Building).\
n\nAbstract\nI will present a new method to compute the parity of the rank
of an elliptic curve and will comment on how this construction generalise
s to Jacobians of curves. This method involves studying the local arithmet
ic attached to covers of the curve. In addition\, I will discuss applicati
ons to the Birch and Swinnerton-Dyer conjecture\, including a new proof of
the parity conjecture for elliptic curves. This is joint work with Vladim
ir Dokchitser\, Alexandros Konstantinou\, Céline Maistret and Adam Morgan
.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Mudgal (University of Oxford)
DTSTART:20230118T160000Z
DTEND:20230118T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/87
DESCRIPTION:Title: Additive equations over lattice points on spheres\nby Akshat Mudg
al (University of Oxford) as part of London number theory seminar\n\nLectu
re held in Rm. 706\, UCL Department of Mathematics (UCL Union Building).\n
\nAbstract\nIn this talk\, we will consider additive properties of lattice
points on spheres. Thus\, defining $S_m$ to be the set of lattice points
on the sphere $x^2 + y^2 + z^2 + w^2 = m$\, we are interested in counting
the number of solutions to the equation\n$$a_1 + a_2 = a_3 + a_4\,$$\nwher
e $a_1\,\\dots\, a_4$ lie in some arbitrary subset $A$ of $S_m$. Such an i
nquiry is closely related to various problems in harmonic analysis and ana
lytic number theory\, including Bourgain's discrete restriction conjecture
for spheres. We will survey some recent results in this direction\, as we
ll as describe some of the various techniques\, arising from areas such as
incidence geometry\, analytic number theory and arithmetic combinatorics\
, that have been employed to tackle this type of problem.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Funke (Durham University)
DTSTART:20230125T160000Z
DTEND:20230125T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/88
DESCRIPTION:Title: Indefinite theta series via incomplete theta integrals\nby Jens F
unke (Durham University) as part of London number theory seminar\n\nLectur
e held in Rm. 706\, UCL Department of Mathematics (UCL Union Building).\n\
nAbstract\nPositive definite theta series have been a classical tool in th
e arithmetic of quadratic forms and also in the theory of modular forms. I
n comparison\, the indefinite case has been less studied. \nIn this talk w
e will explain how indefinite theta series naturally arise in the context
of symmetric spaces of orthogonal type and discuss recent developments ins
pired by mathematical physics. This is joint work with Steve Kudla.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Min Lee (University of Bristol)
DTSTART:20230201T160000Z
DTEND:20230201T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/89
DESCRIPTION:Title: An extension of converse theorems to the Selberg class\nby Min Le
e (University of Bristol) as part of London number theory seminar\n\nLectu
re held in Rm. 706\, UCL Department of Mathematics (UCL Union Building).\n
\nAbstract\nThe converse theorem for automorphic forms has a long history
beginning with the work of Hecke (1936) and a work of Weil (1967): relatin
g the automorphy relations satisfied by classical modular forms to analyti
c properties of their L-functions and the L-functions twisted by Dirichlet
characters. The classical converse theorems were reformulated and general
ised in the setting of automorphic representations for GL(2) by Jacquet an
d Langlands (1970). Since then\, the converse theorem has been a cornersto
ne of the theory of automorphic representations. \n\nVenkatesh (2002)\, in
his thesis\, gave new proof of the classical converse theorem for modular
forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. I
n this talk\, we extend Venkatesh’s proof of the converse theorem to for
ms of arbitrary levels and characters with the gamma factors of the Selber
g class type. \n\n\nThis is joint work with Andrew R. Booker and Michael F
armer.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pak-Hin Lee (University of Leicester and University of Warwick)
DTSTART:20230301T160000Z
DTEND:20230301T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/90
DESCRIPTION:Title: On the p-adic interpolation of Asai L-values\nby Pak-Hin Lee (Uni
versity of Leicester and University of Warwick) as part of London number t
heory seminar\n\nLecture held in Rm. 706\, UCL Department of Mathematics (
UCL Union Building).\n\nAbstract\nOne theme of the relative Langlands prog
ram is that period integrals of an automorphic representation of $G$ over
a subgroup $H$ often detect functorial transfer from some other group $G'$
\; moreover\, such period integrals often compute special $L$-values. It i
s natural to expect $p$-adic $L$-functions interpolating these period inte
grals as the automorphic representation varies in $p$-adic families\, whic
h should encode geometric information about the eigenvariety of $G$. In th
is talk\, we consider the case of Flicker--Rallis periods\, where $G = \\m
athrm{GL}_n(K)$ and $H = \\mathrm{GL}_n(\\mathbf{Q})$ for an imaginary qua
dratic field $K$\, and outline the construction of a $p$-adic $L$-function
on the eigenvariety of $G$ interpolating certain non-critical Asai $L$-va
lues. This is work in progress with Daniel Barrera Salazar and Chris Willi
ams.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Arpin (Leiden University)
DTSTART:20230315T160000Z
DTEND:20230315T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/91
DESCRIPTION:Title: Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs
\nby Sarah Arpin (Leiden University) as part of London number theory s
eminar\n\nLecture held in Rm. 706\, UCL Department of Mathematics (UCL Uni
on Building).\n\nAbstract\nThe classical Deuring correspondence provides a
roadmap between supersingular elliptic curves and the maximal orders whic
h are isomorphic to their endomorphism rings. Building on this idea\, we a
dd the information of a cyclic subgroup of prime order $N$ to supersingula
r elliptic curves\, and prove a generalisation of the Deuring corresponden
ce for these objects. We also study the resulting $\\ell$-isogeny graphs s
upersingular elliptic curve with level-$N$ structure\, and the correspondi
ng graphs in the realm of quaternion algebras. The structure of the supers
ingular elliptic curve ell-isogeny graph underlies the security of a new c
ryptographic signature protocol\, SQISign\, which is proposed to be resist
ant against both classical and quantum attack.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hung Bui (University of Manchester)
DTSTART:20230322T160000Z
DTEND:20230322T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/92
DESCRIPTION:Title: EGGS\, squares and integer points on hyperelliptic curves\nby Hun
g Bui (University of Manchester) as part of London number theory seminar\n
\nLecture held in Rm. 706\, UCL Department of Mathematics (UCL Union Build
ing).\n\nAbstract\nErdos\, Graham and Selfridge considered\, for each posi
tive integer n\, the least value of $t_n$ so that the integers $n+1\, n+2\
,\\ldots\, n+t_n$ contain a subset the product of whose members with $n$ i
s a square. An open problem posed by Granville concerns the size of $t_n$
under the assumption of the ABC Conjecture. We discuss recent work\, joint
with Kyle Pratt and Alexandru Zaharescu\, in which we establish some resu
lts on the distribution of $t_n$\, including an unconditional resolution o
f Granville's problem.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lassina Dembélé (King's College London)
DTSTART:20230222T160000Z
DTEND:20230222T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/93
DESCRIPTION:Title: Congruences of Hilbert-Siegel modular forms and applications.\nby
Lassina Dembélé (King's College London) as part of London number theory
seminar\n\nLecture held in Roberts Building G08\, Sir David Davies LT.\n\
nAbstract\nIn this talk\, we will explain how to compute congruences of Hi
lbert-Siegel modular forms using compact inner forms of ${\\rm GSp}_{2g}$
given by unitary quaternionic groups. We will then give several applicatio
ns of our algorithms.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Tyrell (University of Oxford)
DTSTART:20230208T160000Z
DTEND:20230208T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/94
DESCRIPTION:Title: Further afield and further\, a field: remarks on undecidability\n
by Brian Tyrell (University of Oxford) as part of London number theory sem
inar\n\nLecture held in Rm. 706\, UCL Department of Mathematics (UCL Union
Building).\n\nAbstract\nGiven a field $K$\, one can ask "what first-order
sentences are true in $K$"? E.g. for $K = \\mathbb{C}$\, "$\\exists x (x^
2 = -1)$" is true\, but for $K = \\mathbb{Q}$ this is false. One major are
a of study at the intersection of logic and number theory is\, given a fie
ld $K$ of number-theoretic interest\, whether there is an algorithmic proc
ess which can decide the truth or falsity of a given first-order sentence
in $K$. For $K = \\mathbb{C}$\, there exists such an algorithmic process\;
for $K = \\mathbb{Q}$ there cannot (due to work of Gödel & Julia Robinso
n).\n\nI will pose a related question: whether the logical consequences of
a given sentence in a field may be decided algorithmically. Often the ans
wer is no\; so e.g. we cannot algorithmically detect general properties of
fields $K$ with a Galois extension $L$ such that $\\mathrm{Gal}(L/K) \\co
ng S_5$\, or e.g. general properties of characteristic $p$ fields that adm
it points on a given rationally parameterisable curve over $\\mathbb{F}_p$
. I will focus on those fields whose behaviour is tightly controlled by th
eir absolute Galois group\, and prove some precise limitations.\n\nI will
aim for this talk to be self-contained on the logic side of things!\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Félicien Comtat (Queen Mary University of London)
DTSTART:20230308T160000Z
DTEND:20230308T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/95
DESCRIPTION:Title: Weighted vertical equidistribution of Satake parameters for GSp(4)\nby Félicien Comtat (Queen Mary University of London) as part of London
number theory seminar\n\nLecture held in Rm. 706\, UCL Department of Math
ematics (UCL Union Building).\n\nAbstract\nThe automorphic representations
of an algebraic group G factor as a restricted tensor product of local re
presentations. In turn\, these local representations are parametrised by t
heir Satake parameters. One can ask what properties the Satake parameters
that do arise from automorphic representations of G have to satisfy. For i
nstance\, in the verical distribution problem\, one fixes a prime p and as
ks for the distribution of the Satake parameters at p of automorphic repre
sentations of G varying in some families amenable to the theory of (relati
ve) trace formulae. In this talk\, I discuss the case of Maass forms on G=
GSp(4). When counted with a suitable weight coming from the Kuznetsov form
ula\, the Satake parameters equidistribute with respect to the Sato-Tate m
easure. This is consistent with the generalised Ramanujan conjecture\, exp
ected to hold in this situation.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oli Gregory (Imperial College London)
DTSTART:20230426T150000Z
DTEND:20230426T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/96
DESCRIPTION:Title: A semistable variational p-adic Hodge conjecture\nby Oli Gregory
(Imperial College London) as part of London number theory seminar\n\nLectu
re held in KCL\, Strand Building\, Room S3.30.\n\nAbstract\nLet $k$ be a p
erfect field of characteristic $p>0$\, and let $X$ be a proper scheme over
$W(k)$ with semistable reduction. I shall formulate an analogue of the Fo
ntaine-Messing variational p-adic Hodge conjecture in this setting. To get
there\, I shall define a logarithmic version of motivic cohomology for th
e special fibre $X_k$. This theory is related to relative log-Milnor K-the
ory\, logarithmic Hyodo-Kato Hodge-Witt cohomology\, and log K-theory. Wit
h this in hand\, I shall prove the deformational part of the conjecture\,
simultaneously generalising the semistable $p$-adic Lefschetz $(1\,1)$ the
orem of Yamashita (the case $r=1$) and the deformational $p$-adic Hodge co
njecture of Bloch-Esnault-Kerz (the good reduction case). This is joint wo
rk with Andreas Langer.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Stadlmann (University of Oxford)
DTSTART:20230503T150000Z
DTEND:20230503T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/97
DESCRIPTION:Title: The mean square gap between primes\nby Julia Stadlmann (Universit
y of Oxford) as part of London number theory seminar\n\nLecture held in KC
L\, Strand Building\, Room S3.30.\n\nAbstract\nConditional on the Riemann
hypothesis\, Selberg showed in 1943 that the average size of the squares o
f differences between consecutive primes less than $x$ is $O(log(x)^4)$. U
nconditional results still fall far short of this conjectured bound: Peck
gave a bound of $O(x^{0.25+\\epsilon})$ in 1996 and to date this is the be
st known bound obtained using only methods from classical analytic number
theory.\n\n\nIn this talk we discuss how sieve theory (in the form of Harm
an's sieve) can be combined with classical methods to improve bounds on th
e number of short intervals which contain no primes\, thus improving the u
nconditional bound on the mean square gap between primes to $O(x^{0.23+\\e
psilon})$.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Daw (University of Reading)
DTSTART:20230510T150000Z
DTEND:20230510T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/98
DESCRIPTION:Title: Large Galois orbits for unlikely intersections\nby Christopher Da
w (University of Reading) as part of London number theory seminar\n\nLectu
re held in KCL\, Strand Building\, Room S3.30.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lorenzo La Porta (King's College London)
DTSTART:20230517T150000Z
DTEND:20230517T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/99
DESCRIPTION:Title: Generalised theta operators\nby Lorenzo La Porta (King's College
London) as part of London number theory seminar\n\nLecture held in KCL\, S
trand Building\, Room S3.30.\n\nAbstract\nThe study of the classical theta
operator was key to Edixhoven's proof of the weight part of Serre's modul
arity conjecture. Since then\, a lot of work has been devoted to extending
the construction of this operator to other Shimura varieties\, with an ey
e towards generalisations of Serre's conjecture.\nMy goal is to give an ov
erview of a family of generalised theta operators\, on certain unitary Shi
mura varieties\, that I constructed in my thesis and studied in subsequent
work.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ross Paterson (University of Bristol)
DTSTART:20230524T150000Z
DTEND:20230524T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/100
DESCRIPTION:Title: Quadratic Twists as Random Variables\nby Ross Paterson (Universi
ty of Bristol) as part of London number theory seminar\n\nLecture held in
KCL\, Strand Building\, Room S3.30.\n\nAbstract\nIf $E/\\mathbb{Q}$ is an
elliptic curve\, and $d$ is a squarefree integer\, then the $2$-torsion mo
dules of $E$ and its quadratic twist $E_d$ are isomorphic. In particular t
heir $2$-Selmer groups can be made to lie in the same space. Poonen-Rains
provide a heuristic model for the behaviour of these $2$-Selmer groups ind
ividually\, as E varies\, but how independent are they? We'll present resu
lts in this direction.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucile Devin (Université du Littoral Côte d'Opale)
DTSTART:20230531T150000Z
DTEND:20230531T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/101
DESCRIPTION:Title: Exceptional biases (in the distribution of irreducible polynomials o
ver finite fields)\nby Lucile Devin (Université du Littoral Côte d'O
pale) as part of London number theory seminar\n\nLecture held in KCL\, Str
and Building\, Room S3.30.\n\nAbstract\nStudying the secondary terms of th
e Prime Number Theorem in Arithmetic Progressions\, Chebyshev claimed that
there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4.
This claim was later corrected by Littlewood\, explained\, and quantified
by Rubinstein and Sarnak.\nPursuing the work of Cha\, we investigate analo
gues to Chebyshev's bias in the setting of irreducible polynomials over fi
nite fields. In particular\, we observe exceptional behaviors occurring wh
en the zeros of the involved L-functions are not linearly independent. Mor
e precisely\, we will present instances of "complete bias" and "reversed b
ias"\, and explain why they occur with probability tending to 0\, in the l
arge q limit.\n\nThis is joint work with Bailleul\, Keliher and Li.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jesse Pajwani (Imperial College London)
DTSTART:20230607T150000Z
DTEND:20230607T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/102
DESCRIPTION:Title: The valuative section conjecture and étale homotopy\nby Jesse P
ajwani (Imperial College London) as part of London number theory seminar\n
\nLecture held in KCL\, Strand Building\, Room S3.30.\n\nAbstract\nThe p-a
dic section conjecture is a long standing conjecture of Grothendieck about
curves of high genus over p-adic fields\, linking the p-adic points of a
curve to sections of a short exact sequence of étale fundamental groups.
A powerful way of interpreting the section conjecture is as a fixed point
statement\, and this interpretation makes the statement look like many oth
er theorems in algebraic topology. For this talk\, we'll first introduce t
he framing of the section conjecture as a fixed point statement\, and then
show this interpretation allows us to give an alternate proof of part of
a result of Pop and Stix towards the section conjecture. This new proof ge
neralises to other fields\, and the new fields allow us to extend the orig
inal result to a larger class of varieties.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No talk - London-Paris Number Theory Seminar
DTSTART:20230614T150000Z
DTEND:20230614T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/103
DESCRIPTION:by No talk - London-Paris Number Theory Seminar as part of Lon
don number theory seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cathy Swaenepol (Institut de Mathématiques de Jussieu-Paris Rive
Gauche)
DTSTART:20230621T150000Z
DTEND:20230621T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/104
DESCRIPTION:Title: Primes and squares with preassigned digits\nby Cathy Swaenepol (
Institut de Mathématiques de Jussieu-Paris Rive Gauche) as part of London
number theory seminar\n\nLecture held in KCL\, Strand Building\, Room S3.
30.\n\nAbstract\nBourgain (2015) estimated the number of prime numbers wit
h a positive\nproportion of preassigned digits in base 2. We first presen
t a\ngeneralization of this result to any base $g\\geq 2$. We then discus
s\na more recent result for the set of squares\, which may be seen as one\
nof the most interesting sets after primes. More precisely\, for any\nbas
e $g\\geq 2$\, we obtain an asymptotic formula for the number of\nsquares
with a proportion $c>0$ of preassigned digits. Moreover we\nprovide explic
it admissible values for $c$ depending on $g$. Our\nproof mainly follows
the strategy developed by Bourgain for primes in\nbase 2\, with new diffic
ulties for squares. It is based on the circle\nmethod and combines techniq
ues from harmonic analysis together with\narithmetic properties of squares
and bounds for quadratic Weyl sums.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ollie McGrath (King's College London)
DTSTART:20230628T150000Z
DTEND:20230628T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/105
DESCRIPTION:Title: The asymmetric additive energy of polynomials\nby Ollie McGrath
(King's College London) as part of London number theory seminar\n\nLecture
held in KCL\, Strand Building\, Room S3.30.\n\nAbstract\nIn this talk we
will see how sieve techniques can be used to count the number of solutions
to certain Diophantine equations and in particular prove that polynomials
have small "asymmetric additive energy."\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Otto Overkamp (Heinrich-Heine-Universität Düsseldorf)
DTSTART:20231011T150000Z
DTEND:20231011T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/106
DESCRIPTION:Title: A proof of Chai’s conjecture\nby Otto Overkamp (Heinrich-Heine
-Universität Düsseldorf) as part of London number theory seminar\n\nLect
ure held in Room 140\, the Huxley Building\, Imperial College London.\n\nA
bstract\nThe base change conductor is an invariant which measures the fail
ure of a semiabelian variety to have semiabelian reduction. It was conject
ured by Chai that this invariant is additive in certain exact sequences. I
shall report on recent joint work with Takashi Suzuki which implies this
conjecture. Time permitting\, I shall also discuss counterexamples to a ge
neralisation of Chai’s conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julian Demeio (Bath)
DTSTART:20231018T150000Z
DTEND:20231018T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/107
DESCRIPTION:Title: Hilbert Property for Hilbert modular surfaces\nby Julian Demeio
(Bath) as part of London number theory seminar\n\nLecture held in Room 140
\, the Huxley Building\, Imperial College London.\n\nAbstract\nWork in pro
gress with Damián Gvirtz. We prove the Hilbert Property for several Hilbe
rt modular surfaces. Some of these are K3s\, and the elliptic fibration me
thod is employed. As an application\, we obtain a positive answer to the i
nverse Galois problem in some new cases: namely for $PSL_2(F_{p^2})$ for $
p$ lying in a union of several arithmetic sequences.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amina Abdurrahman (IHES)
DTSTART:20231115T160000Z
DTEND:20231115T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/108
DESCRIPTION:Title: Square roots of symplectic L-functions and Reidemeister torsion\
nby Amina Abdurrahman (IHES) as part of London number theory seminar\n\nLe
cture held in Room 140\, the Huxley Building\, Imperial College London.\n\
nAbstract\nWe give a purely topological formula for the square class of th
e central value of the L-function of a symplectic representation on a curv
e. We also formulate a topological analogue of the statement\, in which th
e central value of the L-function is replaced by Reidemeister torsion of 3
-manifolds. This is related to the theory of epsilon factors in number the
ory and Meyer's signature formula in topology among other topics. We will
present some of these ideas and sketch aspects of the proof. This is joint
work with Akshay Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco D’Addezio (Jussieu)
DTSTART:20231122T160000Z
DTEND:20231122T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/109
DESCRIPTION:Title: The p-torsion of the Brauer group of an abelian variety\nby Marc
o D’Addezio (Jussieu) as part of London number theory seminar\n\nLecture
held in Room 140\, the Huxley Building\, Imperial College London.\n\nAbst
ract\nI will present a new finiteness result for the $p$-primary torsion o
f the transcendental Brauer group of abelian varieties in characteristic $
p$. This follows from a certain “fppf variant” of the Tate conjecture
for abelian varieties. The main ingredient in the proof is de Jong's cryst
alline Tate conjecture. In the talk\, I will recall de Jong's theorem\, th
e relation between crystalline cohomology and the fppf cohomology of $\\mu
_p^n$\, and I will explain some steps of the proof.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (Columbia)
DTSTART:20231129T160000Z
DTEND:20231129T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/110
DESCRIPTION:Title: Hecke algebras for p-adic groups and explicit Local Langlands Corres
pondence\nby Yujie Xu (Columbia) as part of London number theory semin
ar\n\nLecture held in Room 140\, the Huxley Building\, Imperial College Lo
ndon.\n\nAbstract\nI will talk about several results on Hecke algebras att
ached to Bernstein blocks of (arbitrary) reductive p-adic groups\, where w
e construct a local Langlands correspondence for these Bernstein blocks. O
ur techniques draw inspirations from the foundational works of Deligne\, K
azhdan and Lusztig. \n\nAs an application\, we prove the (classical) Local
Langlands Conjecture for G_2\, which is the first known case in literatur
e of (classical) LLC for exceptional groups. Our correspondence satisfies
an expected property on cuspidal support\, which is compatible with the ge
neralized Springer correspondence (for Lusztig's perverse sheaves)\, along
with a list of characterizing properties including the stabilization of c
haracter sums. In particular\, we obtain "mixed" L-packets containing "F-s
ingular" supercuspidals and non-supercuspidals. Such "mixed" L-packets had
been elusive up until this point and very little was known prior to our w
ork. I will give explicit examples of such mixed L-packets using Deligne-L
usztig theory and Kazhdan-Lusztig parametrization. \n\nIf time permits\, I
will explain how to pin down certain choices in the construction of the c
orrespondence using stability of L-packets\; one key input is a homogeneit
y result due to Waldspurger and DeBacker. Moreover\, I will mention how to
adapt our general strategy to construct explicit LLC for other reductive
groups\, such as GSp(4)\, Sp(4)\, etc. Such explicit description of the L-
packets (e.g. the Kazhdan-Lusztig parameters) has been useful in applicati
ons to modularity lifting questions as in the recent work of Whitmore. \n\
nSome parts of this talk are based on my joint work with Aubert\, and some
other parts are based on my joint work with Suzuki.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pol van Hoften (VU Amsterdam)
DTSTART:20231206T160000Z
DTEND:20231206T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/111
DESCRIPTION:Title: On Exotic Hecke correspondences\nby Pol van Hoften (VU Amsterdam
) as part of London number theory seminar\n\nLecture held in Room 140\, th
e Huxley Building\, Imperial College London.\n\nAbstract\nThe goal of this
talk is to explain joint work in progress with Jack Sempliner on the cons
truction of "exotic" Hecke correspondences between the mod p fibers of dif
ferent Shimura varieties of Hodge type. Our work generalizes forthcoming w
ork of Xiao-Zhu\; our results cover the new situation where the groups und
erlying the two different Shimura varieties are allowed to be to be non-is
omorphic at p. As a consequence of our main results\, we obtain exotic iso
morphisms of Igusa varieties in the style of Caraiani-Tamiozzo. In the tal
k\, I will give many illustrative examples.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emiliano Ambrosi (Strasbourg)
DTSTART:20231213T160000Z
DTEND:20231213T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/112
DESCRIPTION:Title: Reduction modulo p of the Noether problem\nby Emiliano Ambrosi (
Strasbourg) as part of London number theory seminar\n\nLecture held in Roo
m 140\, the Huxley Building\, Imperial College London.\n\nAbstract\nLet k
be an algebraically closed field of characteristic p≥0 and V a faithful
k-rational representation of an l-group G. The Noether's problem asks whet
her V/G is (stably) birational to a point. If l is equal to p\, then Kuniy
oshi proved that this is true\, while\, if l is different from p\, Saltman
constructed l-groups for which V/G is not stably rational. Hence\, the ge
ometry of V/G depends heavily on the characteristic of the field. We sho
w that for all the groups G constructed by Saltman\, one cannot interpolat
e between the Noether problem in characteristic 0 and p. More precisely\,
we show that it does not exist a complete valuation ring R of mixed charac
teristic (0\,p) and a smooth proper R-scheme X---->Spec(R) whose special f
iber and generic fiber are both stably birational to V/G. The proof combin
es the integral p-adic Hodge theoretic results of Bhatt-Morrow-Scholze\, w
ith the study of the Cartier operator on differential forms in positive ch
aracteristic. This is a joint work with Domenico Valloni.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wataru Kai (Tohoku University)
DTSTART:20231108T160000Z
DTEND:20231108T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/113
DESCRIPTION:Title: Linear patterns of prime elements in number fields\nby Wataru Ka
i (Tohoku University) as part of London number theory seminar\n\nLecture h
eld in Room 140\, the Huxley Building\, Imperial College London.\n\nAbstra
ct\nI will discuss my recent result that gives a sufficient condition for
a set of finitely many polynomials of degree 1 with coefficients in a numb
er ring to attain simultaneous prime values. This extends a 2012 theorem o
f Green-Tao-Ziegler from the case of Z to the general case. Time permittin
g\, I will mention how this can be applied to produce (modestly) new famil
ies of varieties over number fields which satisfy the Hasse principle for
rational points by using the so-called fibration methods.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ambrus Pal (Imperial)
DTSTART:20231025T150000Z
DTEND:20231025T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/114
DESCRIPTION:Title: Tate's conjecture for certain Drinfeld modular surfaces\nby Ambr
us Pal (Imperial) as part of London number theory seminar\n\nLecture held
in Room 140\, the Huxley Building\, Imperial College London.\n\nAbstract\n
Tate's conjecture on algebraic cycles is one of the central conjectures in
arithmetic geometry\, but it is open even for codimension one cycles. The
re are only a few classes of varieties when this claim is known. I will re
port about a new class of surfaces defined over global function fields whi
ch were defined by Stuhler in there original form\, and are closely analog
ous to Hilbert modular surfaces. Our proof employs p-adic methods\, includ
ing the p-adic Lefschetz (1\,1) theorem proved by Lazda and myself. We als
o exploit that these varieties are totally degenerate in the sense of Rask
ind\, but this is not sufficient\, we need some information from the Langl
ands correspondence\, too. I will also talk about some particularly simple
surfaces which show that the first property cannot be used to give a quic
ker proof. Joint work with Koskivirta.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bence Hevesi (Imperial)
DTSTART:20231101T160000Z
DTEND:20231101T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/115
DESCRIPTION:Title: Local-global compatibility at l=p for torsion automorphic Galois rep
resentations\nby Bence Hevesi (Imperial) as part of London number theo
ry seminar\n\nLecture held in Room 140\, the Huxley Building\, Imperial Co
llege London.\n\nAbstract\nSome 10 years ago\, Scholze proved the existenc
e of Galois representations associated with torsion classes appearing in t
he cohomology of locally symmetric spaces for GL_n over imaginary CM field
s. Since then\, the question of local-global compatibility for his automor
phic Galois representations has been an active area of research. I will re
port on my work on verifying a rather general local-global compatibility a
t l=p in this direction\, generalising the already existing results of the
celebrated 10 author paper and Caraiani—Newton.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Honnor (Imperial)
DTSTART:20240110T160000Z
DTEND:20240110T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/116
DESCRIPTION:Title: Refined conjectures of Birch and Swinnerton-Dyer type\nby Matthe
w Honnor (Imperial) as part of London number theory seminar\n\nLecture hel
d in Chadwick G07 (UCL).\n\nAbstract\nIn analogy to Stickelberger's Theore
m and refining the Birch—Swinnerton-Dyer Conjecture\, Mazur—Tate conje
cture an order of vanishing and main conjecture for a certain group ring e
lement. This element is defined in terms of modular symbols and relates t
o the twisted Hasse—Weil $L$-series of elliptic curves. In this talk I w
ill explain the conjectures of Mazur—Tate and report on work in progress
\, joint with Dominik Bullach\, in which we prove new results towards thes
e conjectures.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dominik Bullach (UCL)
DTSTART:20240117T160000Z
DTEND:20240117T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/117
DESCRIPTION:Title: On some recent developments in the theory of Euler systems\nby D
ominik Bullach (UCL) as part of London number theory seminar\n\nLecture he
ld in G02 Watson LT\, Medawar Building.\n\nAbstract\nEver since their intr
oduction\, Euler systems have played an important role in \nproving conjec
tures on leading terms of L-series such as instances of the Tamagawa \nNum
ber Conjecture of Bloch and Kato. In this talk\, I will survey some recent
developments \nin the general theory of Euler systems\, including joint w
ork in progress with David Burns.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Netan Dogra (KCL)
DTSTART:20240124T160000Z
DTEND:20240124T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/118
DESCRIPTION:Title: On the Zilber-Pink conjecture for a product of curves\nby Netan
Dogra (KCL) as part of London number theory seminar\n\nLecture held in Exe
cutive Suite 103\, Engineering Front Building.\n\nAbstract\nLet $X$ be a c
urve of genus $g>1$ over the complex numbers. What is the Zariski closure\
, inside $X^n$\, of the set of $n$-tuples of points $(z_i)$ for which ther
e exists a non-constant function $f$ on $X$ with divisor supported on $\\{
z_i\\}$? This question can be viewed as a special case of the Zilber-Pink
conjecture\, which is a broad generalisation of the Andre-Oort conjecture.
In this talk I will describe new results which answer this question for s
ome $(X\,n)$. This is joint work with Arnab Saha (IIT Gandhinagar).\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ju-Feng Wu (University of Warwick)
DTSTART:20240131T160000Z
DTEND:20240131T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/119
DESCRIPTION:Title: Overconvergent Eichler—Shimura morphisms for Siegel modular forms<
/a>\nby Ju-Feng Wu (University of Warwick) as part of London number theory
seminar\n\nLecture held in 9 Garwood LT\, South Wing.\n\nAbstract\nA theo
rem of Faltings—Chai provides a comparison between the étale cohomology
of the Siegel modular variety and the coherent cohomology of automorphic
sheaves\, which generalises the classical Eichler—Shimura decomposition
in the case of modular forms. In this talk\, based on joint work with Hans
heng Diao and Giovanni Rosso\, I will discuss how to $p$-adically interpol
ate the result of Faltings—Chai. The strategy is inspired by the work of
Chojecki—Hansen—Johansson\; one of the key ingredients is Higher Cole
man Theory\, recently introduced by Boxer—Pilloni.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cecilia Busuioc (UCL)
DTSTART:20240207T160000Z
DTEND:20240207T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/120
DESCRIPTION:Title: Modular symbols with values in Beilinson-Kato distributions\nby
Cecilia Busuioc (UCL) as part of London number theory seminar\n\nLecture h
eld in Executive Suite 103\, Engineering Front Building.\n\nAbstract\nIn t
his talk\, we will describe the construction of a $\\operatorname{GL}_n(\\
mathbb{Q})$-invariant modular symbol with coefficients in a space of distr
ibutions that take values in Milnor K-groups of modular function fields.
This is based on joint work with J. Park\, O. Patashnick and G. Stevens.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Newton (Oxford)
DTSTART:20240221T160000Z
DTEND:20240221T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/121
DESCRIPTION:Title: Base change for modular forms\nby James Newton (Oxford) as part
of London number theory seminar\n\nLecture held in Executive Suite 103\, E
ngineering Front Building.\n\nAbstract\nI'll talk about the base change li
fting from holomorphic modular forms to Hilbert modular forms for totally
real fields F. A new proof of the existence of this base change lifting is
contained in joint work with Laurent Clozel and Jack Thorne. \n\nThe base
change lifting is a simple example of Langlands functoriality\, correspon
ding on the Galois side to restriction to the absolute Galois group of F.
When F is a solvable extension of Q\, its existence was proved by Langland
s using the twisted trace formula (earlier work by Doi and Naganuma covere
d the case where F is quadratic). Dieulefait used modularity lifting theor
ems and a delicate construction of chains of congruences between modular f
orms to prove the existence of the base change lifting without a solvabili
ty assumption. Our new proof replaces (at least some of) this chain of con
gruences with a `p-adic analytic continuation of functoriality' step\, ada
pted from my work with Thorne on symmetric power functoriality.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asbjørn Nordentoft (Université Paris-Saclay)
DTSTART:20240228T160000Z
DTEND:20240228T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/122
DESCRIPTION:Title: Horizontal p-adic L-functions\nby Asbjørn Nordentoft (Universit
é Paris-Saclay) as part of London number theory seminar\n\nLecture held i
n A1/3\, Physics Building.\n\nAbstract\nGoldfeld’s Conjecture predicts t
hat exactly 50% of quadratic twists of a fixed elliptic curve will have L-
function vanishing at the central point. When considering the non-vanishin
g of twists of elliptic curve L-functions by characters of (fixed) order g
reater than 2\, it has been predicted by David-Fearnly-Kisilevsky that 100
% should be non-vanishing. Very little was previously known outside the qu
adratic case as the problem lies beyond the current technology of e.g. ana
lytic number theory. In this talk I will present a p-adic approach relying
on the construction of a ‘horizontal p-adic L-function’. This approac
h yields strong quantitative non-vanishing results for general order twist
s. In particular\, we obtain the best bound towards Goldfeld's Conjecture
for one hundred percent of elliptic curves (improving on a result of Ono).
\n\nThis is joint work with Daniel Kriz.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shaun Stevens (University of East Anglia)
DTSTART:20240306T160000Z
DTEND:20240306T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/123
DESCRIPTION:Title: How many local factors does it take to determine a representation?\nby Shaun Stevens (University of East Anglia) as part of London number
theory seminar\n\nLecture held in Medawar G02 Watson.\n\nAbstract\nIn 1993
\, Henniart proved that the (then conjectural) Local Langlands Corresponde
nce for $\\operatorname{GL}(n$) is determined by gamma-factors of pairs. M
ore precisely\, he proved a local converse theorem: for $F$ a non-archimed
ean local field\, an irreducible (smooth complex) representation $\\pi$ of
$\\operatorname{GL}(n\,F)$ is determined by the collection of gamma-facto
rs of the pairs $(\\pi\,\\tau)$ as $\\tau$ runs through the irreducible re
presentations of all $\\operatorname{GL}(m\,F)$ with $m < n$. More recentl
y\, Chai (2019) and Jacquet—Liu (2018) showed that one only needs to con
sider $m\\leq n/2$ to determine $\\pi$. This bound on $m$ is best possible
\, at least when $n$ is less that the residual characteristic of $F$\, by
work of Adrian—Liu—Stevens—Tam (2018) and Adrian (2023). \n\n \n\nFo
r groups other than $\\operatorname{GL}(n)$ there are additional complicat
ions. I’ll explain what I know is already known about this problem and r
eport on some joint work with Moshe Adrian\, which gives some answers but
also leaves many questions.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Fisher (Cambridge)
DTSTART:20240313T160000Z
DTEND:20240313T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/124
DESCRIPTION:Title: Computing the Cassels-Tate pairing on the 2-Selmer group of a genus
2 Jacobian\nby Tom Fisher (Cambridge) as part of London number theory
seminar\n\nLecture held in Medawar G02 Watson.\n\nAbstract\nIn her 2021 th
esis\, my student Jiali Yan gave a practical method\nfor computing the Cas
sels-Tate pairing on the 2-Selmer group of the\nJacobian of a genus 2 curv
e all of whose Weierstrass points are rational.\nShe also gave a second me
thod without any assumption on the Weierstrass\npoints\, but instead assum
ing we can find a rational point on a certain\ntwisted Kummer surface. The
two methods can be thought of as generalising\nmethods of Cassels and Don
nelly in the elliptic curve case. I will\ndescribe a practical refinement
of the second method which is now\nimplemented in Magma\, and has been use
d to unconditionally determine the\nranks of all genus 2 Jacobians in the
LMFDB.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Wadsley
DTSTART:20240320T160000Z
DTEND:20240320T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/125
DESCRIPTION:Title: Equivariant vector bundles with connection on the p-adic half-plane<
/a>\nby Simon Wadsley as part of London number theory seminar\n\nLecture h
eld in Watson Lecture Theatre G02\, Medawar Building.\n\nAbstract\nRecent
joint work with Konstantin Ardakov has been devoted\nto classifying equiva
riant line bundles with flat connection on the\nDrinfeld $p$-adic half-pla
ne defined over $F$\, a finite extension of $\\mathbb{Q}_p$\,\nand proving
that their global sections yield admissible locally\nanalytic representat
ions of $\\operatorname{GL}_2(F)$ of finite length. In this talk we\nwill
discuss this work and invite reflection on how it might be\nextended to eq
uivariant vector bundles with connection on the $p$-adic\nhalf-plane.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Hauke (University of York)
DTSTART:20240501T150000Z
DTEND:20240501T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/126
DESCRIPTION:Title: Duffin-Schaeffer meets Littlewood and related topics\nby Manuel
Hauke (University of York) as part of London number theory seminar\n\nLect
ure held in Room K0.18 King's Building\, King's College London (Strand Cam
pus).\n\nAbstract\nKhintchine's Theorem is one of the cornerstones in metr
ic Diophantine approximation. The question of removing the monotonicity co
ndition on the approximation function in Khintchine's Theorem led to the r
ecently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue o
f Khintchine's Theorem for multiplicative Diophantine approximation\, agai
n assuming monotonicity. \n \n In this talk\, I will discuss my joint work
with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theor
em. Furthermore\, I will give a broader overview on various questions in m
etric Diophantine approximation and demonstrate the deep connection to ana
lytic number theory that lies in the heart of the corresponding proofs.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Efthymios Sofos (University of Glasgow)
DTSTART:20240508T150000Z
DTEND:20240508T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/127
DESCRIPTION:Title: 6-torsion and integral points on quartic surfaces\nby Efthymios
Sofos (University of Glasgow) as part of London number theory seminar\n\nL
ecture held in Room K0.18 King's Building\, King's College London (Strand
Campus).\n\nAbstract\nI will discuss some new results on averages of multi
plicative functions over integer sequences. We will then give applications
to Cohen-Lenstra and Manin's conjecture. Joint work with Chan\, Koymans a
nd Pagano.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lewis Combes (University of Sheffield)
DTSTART:20240515T150000Z
DTEND:20240515T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/128
DESCRIPTION:Title: Period polynomials of Bianchi modular forms\nby Lewis Combes (Un
iversity of Sheffield) as part of London number theory seminar\n\nLecture
held in Room K0.18 King's Building\, King's College London (Strand Campus)
.\n\nAbstract\nBianchi modular forms (i.e. automorphic forms over imaginar
y quadratic fields) share many similarities with their classical cousins.
One such similarity is the period polynomial\, studied for classical modul
ar forms by Manin\, Kohnen and Zagier\, as well as many others. In this ta
lk we define period polynomials of Bianchi modular forms\, show how to com
pute them in practice\, and use them to (conjecturally) extract informatio
n about congruences between Bianchi forms of various types (base-change an
d genuine forms\; cusp forms and Eisenstein series). All of this is done t
hrough an example space of Bianchi forms\, from which we find new congruen
ces modulo 43 and 173. Time permitting\, we will also describe some open p
roblems relating to these methods\, and how these relate to the classical
picture. No prior knowledge of Bianchi modular forms is assumed.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ofir Gorodetsky (University of Oxford)
DTSTART:20240529T150000Z
DTEND:20240529T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/129
DESCRIPTION:Title: Martingale central limit theorems for weighted sums of random multip
licative functions\nby Ofir Gorodetsky (University of Oxford) as part
of London number theory seminar\n\nLecture held in Room K0.18 King's Build
ing\, King's College London (Strand Campus).\n\nAbstract\nA random multipl
icative function is a multiplicative function alpha(n) such that its value
s on primes\, (alpha(p))_(p=2\,3\,5\,...)\, are i.i.d. random variables. T
he simplest example is the Steinhaus function\, which is a completely mult
iplicative function with alpha(p) uniformly distributed on the unit circle
. A basic question in the field is finding the limiting distribution of th
e (normalized) sum of alpha(n) from n=1 to n=x\, possibly restricted to a
subset of integers of interest. This question is currently resolved only i
n a few cases. We shall describe ongoing work where we are able to find th
e limiting distribution in many new instances of interest. The distributio
n we find is not gaussian\, in contrast to all previous works. This is joi
nt work with Mo Dick Wong (Durham University).\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vandita Patel (University of Manchester)
DTSTART:20240619T150000Z
DTEND:20240619T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/130
DESCRIPTION:Title: Values of the Ramanujan tau-function\nby Vandita Patel (Universi
ty of Manchester) as part of London number theory seminar\n\nLecture held
in KCL K0.18.\n\nAbstract\nThe infamous Ramanujan tau-function is the star
ting point for many mysterious conjectures and difficult open problems wit
hin the realm of modular forms. In this talk\, I will discuss some of our
recent results pertaining to odd values of the Ramanujan tau-function. We
use a combination of tools which include the Primitive Divisor Theorem of
Bilu\, Hanrot and Voutier\, bounds for solutions to Thue–Mahler equation
s due to Bugeaud and Gyory\, and the modular approach via Galois represent
ations of Frey-Hellegouarch elliptic curves. This is joint work with Mike
Bennett (UBC)\, Adela Gherga (Warwick) and Samir Siksek (Warwick)\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mohamed Tawfik (King's College London)
DTSTART:20240424T150000Z
DTEND:20240424T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/131
DESCRIPTION:Title: Brauer-Manin obstructions for Kummer surfaces\nby Mohamed Tawfik
(King's College London) as part of London number theory seminar\n\nLectur
e held in K0.18\, King's Building\, Strand Campus\, King's College London.
\n\nAbstract\nWe start by introducing Brauer-Manin obstructions to local-g
lobal principles over varieties. We then move to some of the known literat
ure on Brauer-Manin obstructions for Kummer surfaces of products of ellipt
ic curves. We finally present our work on some of the special cases where
we calculate the Brauer group of a Kummer surface $X=Kum(E \\times E')$ of
a product of CM elliptic curves $E$ and $E'$\, where $End(E)=End(E')=\\ma
thbb{Z}[\\zeta_3]$\, and show that a non-trivial 5-torsion element of the
transcendental Brauer group gives rise to Brauer Manin obstruction to weak
approximation for $X$.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lazar Radicevic (King's College London)
DTSTART:20240522T150000Z
DTEND:20240522T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/132
DESCRIPTION:Title: Projective geometry and invariant theory of elliptic curves and ring
s of finite rank\nby Lazar Radicevic (King's College London) as part o
f London number theory seminar\n\nLecture held in K0.18\, King's Building\
, Strand Campus\, King's College London.\n\nAbstract\nI will explain how f
ree resolutions of ideals can be used to systematically formulate invarian
t theory for several moduli spaces of varieties that are of interest in ar
ithmetic statistics and computational number theory. In particular\, we ex
tend the classical invariant theory formulas for the Jacobian of a genus o
ne curve of degree n=2\,3\,4\,5 to curves of arbitrary degree\, generalizi
ng the work on genus one models of Cremona\, Fisher and Stoll\, and in a
joint work with Tom Fisher\, we compute structure constants for a rank n r
ing from the free resolution of its associated set of n points in projecti
ve space\, generalizing the previously known constructions of Levi-Delone-
Faddeev and Bhargava. Time permitting I will talk about an ongoing project
to extend these results to abelian varieties of higher dimension.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Baptiste Teyssier (Université Sorbonne)
DTSTART:20240605T150000Z
DTEND:20240605T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/133
DESCRIPTION:Title: Boundedness for Betti numbers of étale sheaves in positive characte
ristic.\nby Jean-Baptiste Teyssier (Université Sorbonne) as part of L
ondon number theory seminar\n\nLecture held in K0.18\, King's Building\, S
trand Campus\, King's College London.\n\nAbstract\nCohomology is the most
fundamental global invariant attached to a sheaf. For a \\bar{Q}_l local s
ystem L on the complement of a divisor D in a smooth projective variety ov
er an algebraically closed field of characteristic p ≠ l\, we will adver
tise the existence of estimates for the rank of each cohomology spaces of
L depending only on local data : the rank of L and the ramification conduc
tors of L at the generic points of D. This is joint work with Haoyu Hu.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sylvy Anscombe (Université Paris Cité)
DTSTART:20240612T150000Z
DTEND:20240612T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/134
DESCRIPTION:Title: Uniform aspects of the theory of complete valued fields\nby Sylv
y Anscombe (Université Paris Cité) as part of London number theory semin
ar\n\nLecture held in K0.18\, King's Building\, Strand Campus\, King's Col
lege London.\n\nAbstract\nA good deal of the arithmetic of a field can be
expressed by sentences in the first-order language of rings. The theories\
nof the characteristic zero local fields have been axiomatized and are dec
idable: in the case of $Q_p$ and its finite extensions\,\nAx\, Kochen\, an
d (independently) Ershov\, gave complete axiomatizations that are centred
on a formalization of Hensel’s\nLemma. In fact the theory of any field o
f characteristic zero which is complete with respect to a non-archimedean\
nvaluation can be likewise axiomatized.\n\nI will explain recent joint wor
k with Jahnke\, and also with Dittmann and Jahnke\, in which we extend the
classical\nwork on these theories to include the case of imperfect residu
e fields. In particular we show that “Hilbert’s Tenth\nProblem” (H10
) in these fields (i.e. the problem of effectively determining whether a g
iven Diophantine equation has\nsolutions) is solvable if and only if the a
nalogous problem is solvable on a structure we define on the residue field
. This\nfollows a pattern of such “transfer” results for H10 — estab
lished for valued fields of positive characteristic in earlier\nwork with
Fehm — although in the current case we really need the extra structure.\
n\nI will describe these results\, focusing on the extent to which they de
pend (or not) on the residue field. If there is\ntime I will discuss the a
forementioned H10 transfer for complete valued fields in positive characte
ristic\, including more\nrecent uniform aspects.\n\nI will not assume a ba
ckground in logic.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ana Caraiani (Imperial College London)
DTSTART:20240626T150000Z
DTEND:20240626T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/135
DESCRIPTION:Title: Towards an Eichler-Shimura decomposition for ordinary p-adic Siegel
modular forms\nby Ana Caraiani (Imperial College London) as part of Lo
ndon number theory seminar\n\nLecture held in K0.18\, King's Building\, St
rand Campus\, King's College London.\n\nAbstract\nThere are two different
ways to construct families of ordinary p-adic Siegel modular forms. One is
by p-adically interpolating classes in Betti cohomology\, first introduce
d by Hida and then given a more representation-theoretic interpretation by
Emerton. The other is by p-adically interpolating classes in coherent coh
omology\, once again pioneered by Hida and generalised in recent years by
Boxer and Pilloni. I will explain these two constructions and then discuss
joint work in progress with James Newton and Juan Esteban Rodríguez Cama
rgo that aims to compare them.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Skinner (Princeton University)
DTSTART:20240605T120000Z
DTEND:20240605T130000Z
DTSTAMP:20260404T111112Z
UID:LNTS/136
DESCRIPTION:Title: Anticyclotomic Euler systems for Conjugate-dual Galois representatio
ns\nby Christopher Skinner (Princeton University) as part of London nu
mber theory seminar\n\nLecture held in K0.18\, King's Building\, Strand Ca
mpus\, King's College London.\n\nAbstract\nI will explain a definition of
Euler systems for anticyclotomic extensions of a CM extension K/F. This al
lows one to prove analogs of Kolyvagin's famous results for Heegner points
(rank one\, finiteness of Tate-Shafarevich groups) for a very general cla
ss of Galois representations over CM fields. A novel feature of this appro
ach is to focus on primes that split in K/F (as opposed to Kolyvagin's ine
rt primes). I will also describe some of the many examples of such Euler
systems that have been constructed recently. This is joint work with Dimit
ar Jetchev and was begun in collaboration with Jan Nekovar.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Wang-Erickson (University of Pittsburgh)
DTSTART:20240605T133000Z
DTEND:20240605T143000Z
DTSTAMP:20260404T111112Z
UID:LNTS/137
DESCRIPTION:Title: Critical Hida theory\, bi-ordinary complexes\, and weight 1 coherent
cohomology\nby Carl Wang-Erickson (University of Pittsburgh) as part
of London number theory seminar\n\nLecture held in K0.18\, King's Building
\, Strand Campus\, King's College London.\n\nAbstract\nColeman made observ
ations about overconvergent modular forms of weight at least 2 and critica
l slope which imply that they are almost spanned by two subspaces correspo
nding to two different kinds of twist of ordinary overconvergent modular f
orms. He also showed that the “almost” is accounted for by a square-ni
lpotent action of Hecke operators. Motivated by questions about Galois rep
resentations associated to these forms\, we intersect these two twists to
define “bi-ordinary” forms. But we do this in a derived way: the sum o
peration from the two twisted ordinary subspaces to the space of critical
forms defines a length 1 “bi-ordinary complex\," making the bi-ordinary
forms the 0th degree of bi-ordinary cohomology and realizing the square-ni
lpotent Hecke action as a degree-shifting action. Relying on classical Hid
a theory as well as the higher Hida theory of Boxer-Pilloni\, we interpola
te this complex over weights. We can deduce “R=T” theorems in the crit
ical and bi-ordinary cases from R=T theorems in the ordinary case. And spe
cializing to weight 1 under a supplemental assumption\, we show that the b
i-ordinary complex with its square-nilpotent Hecke action specializes to w
eight 1 coherent cohomology of the modular curve with a degree-shifting ac
tion of a Stark unit group. The action is a candidate for a p-adic realiza
tion of conjectures about motivic actions of Venkatesh\, Harris\, and Pras
anna. This is joint work with Francesc Castella.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kalyani Kansal (Imperial)
DTSTART:20241009T150000Z
DTEND:20241009T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/138
DESCRIPTION:Title: Non-generic components of the Emerton-Gee stack for GL$_2$\nby K
alyani Kansal (Imperial) as part of London number theory seminar\n\nLectur
e held in Huxley 140\, Imperial College.\n\nAbstract\nLet $K$ be an unrami
fied extension of $\\mathbb{Q}_p$ for a prime p > 3. The reduced part of t
he Emerton-Gee stack for $\\mathrm{GL}_2$ can be viewed as parameterizing
two-dimensional mod p Galois representations of the absolute Galois group
of $K$. In this talk\, we will consider the extremely non-generic irreduci
ble components of this reduced part and see precisely which ones are smoot
h or normal\, and which have Gorenstein or Cohen-Macaulay normalizations\,
as well as determine their singular loci. We will see some consequences o
f this study for the conjectural categorical p-adic Langlands corresponden
ce. This is based on recent joint work with Ben Savoie.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Ortiz (Imperial)
DTSTART:20241016T150000Z
DTEND:20241016T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/139
DESCRIPTION:Title: Theta linkage maps and the weight part of Serre's conjecture\nby
Martin Ortiz (Imperial) as part of London number theory seminar\n\nLectur
e held in Huxley 140\, Imperial College.\n\nAbstract\nThe weight part of S
erre's conjecture seeks to understand mod p congruences of automorphic for
ms of different weights. For modular forms a key ingredient in its proof w
as Edixhoven's use of the theta operator on the modular curve. I will expl
ain the construction of a new family of theta operators on Shimura varieti
es\, and how they are related to the conjectures of Herzig on the weight p
art of Serre's conjecture. As an application I prove a generic entailment
for the group GSp4\, i.e. a Hecke eigenform for a generic Serre weight in
the lowest alcove is also modular for a Serre weight in one of the upper a
lcoves.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seoyoung Kim (University of Göttingen)
DTSTART:20241023T150000Z
DTEND:20241023T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/140
DESCRIPTION:Title: Certain families of K3 surfaces and their modularity\nby Seoyoun
g Kim (University of Göttingen) as part of London number theory seminar\n
\nLecture held in Huxley 140\, Imperial College.\n\nAbstract\nWe start wit
h a double sextic family of K3 surfaces with four parameters with Picard n
umber $16$. Then by geometric reduction (top-to-bottom) processes\, we obt
ain three\, two and one parameter families of K3 surfaces of Picard number
$17\, 18$ and $19$ respectively. All these families turn out to be of hyp
ergeometric type in the sense that their Picard--Fuchs differential equati
ons are given by hypergeometric or Heun functions. We will study the geome
try of two parameter families in detail.\n\nWe will then prove\, after sui
table specializations of parameters\, these K3 surfaces will have CM (com
plex multiplication)\, and will become modular in the sense that the Galoi
s representations of dimensions $\\leq 6$ associated to the transcendental
lattices are all induced from $1$-dimensional representations. Thus\, the
se K3 surfaces will be determined by modular forms of various weights. Thi
s is done starting with one-parameter family establishing the modularity a
t special fibers\, and then applying arithmetic induction (bottom-to-top)
processes to multi-parameter families. This is a joint work with A. Clingh
er\, A. Malmendier\, and N. Yui.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Breuer (University of Newcastle)
DTSTART:20241030T160000Z
DTEND:20241030T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/141
DESCRIPTION:Title: Coefficients of modular polynomials\nby Florian Breuer (Universi
ty of Newcastle) as part of London number theory seminar\n\nLecture held i
n Huxley 140\, Imperial College.\n\nAbstract\nFor a positive integer $N$\,
denote by $\\Phi_N(X\,Y)$ the elliptic modular polynomial of level $N$ --
it vanishes at pairs of $j$-invariants of elliptic curves linked by a cyc
lic isogeny of degree $N$ and plays an important role in various cryptosys
tems. The coefficients of $\\Phi_N$ are notoriously large. In this talk\,
I present joint work with Fabien Pazuki and Desir'ee Gij\\'on G\\'omez pro
ving explicit upper and lower bounds on the size of these coefficients.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akinari Hoshi (Niigata University)
DTSTART:20241106T160000Z
DTEND:20241106T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/142
DESCRIPTION:Title: Norm one tori and Hasse norm principle\nby Akinari Hoshi (Niigat
a University) as part of London number theory seminar\n\nLecture held in H
uxley 140\, Imperial College.\n\nAbstract\nLet $k$ be a field and $T$ be a
n algebraic $k$-torus. In 1969\, over a global field $k$\, Voskresenskii p
roved that there exists an exact sequence $0\\to A(T)\\to H^1(k\,{\\rm Pic
}\\\,\\overline{X})^\\vee\\to Sha(T)\\to 0$ where $A(T)$ is the kernel of
the weak approximation of $T$\, $Sha(T)$ is the Shafarevich-Tate group of
$T$\, $X$ is a smooth $k$-compactification of $T$\, ${\\rm Pic}\\\,\\overl
ine{X}$ is the Picard group of $\\overline{X}=X\\times_k\\overline{k}$ and
$\\vee$ stands for the Pontryagin dual. On the other hand\, in 1963\, Ono
proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ of $K/k$\, $Sha
(T)=0$ if and only if the Hasse norm principle holds for $K/k$. First\, we
determine $H^1(k\,{\\rm Pic}\\\, \\overline{X})$ for algebraic $k$-tori $
T$ up to dimension $5$. Second\, we determine $H^1(k\,{\\rm Pic}\\\, \\ove
rline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ with $[K:k]\\leq 17$.
Third\, we give a necessary and sufficient condition for the Hasse norm pr
inciple for $K/k$ with $[K:k]\\leq 15$. We also show that $H^1(k\,{\\rm Pi
c}\\\, \\overline{X})=0$ or $Z/2Z$ for $T=R^{(1)}_{K/k}(G_m)$ when the Gal
ois group of the Galois closure of $K/k$ is the Mathieu group $M_{11}$ or
the Janko group $J_1$. As applications of the results\, we get the group $
T(k)/R$ of $R$-equivalence classes over a local field $k$ via Colliot-Th\\
'{e}l\\`{e}ne and Sansuc's formula and the Tamagawa number $\\tau(T)$ over
a number field $k$ via Ono's formula $\\tau(T)=|H^1(k\,\\widehat{T})|/|Sh
a(T)|$. This is joint work with Kazuki Kanai and Aiichi Yamasaki.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuma Ohara (MPIM Bonn)
DTSTART:20241113T160000Z
DTEND:20241113T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/143
DESCRIPTION:Title: Reduction to depth zero for tame p-adic groups via Hecke algebra iso
morphisms\nby Kazuma Ohara (MPIM Bonn) as part of London number theory
seminar\n\nLecture held in Huxley 140\, Imperial College.\n\nAbstract\nTh
e category of smooth complex representations of $p$-adic\ngroups decompose
s into a product of indecomposable full subcategories\,\ncalled Bernstein
blocks. In this talk\, I will explain the result that\nunder a mild tamene
ss condition\, every block is equivalent to a depth-zero\nblock\, which is
closely related to the representation theory of finite\nreductive groups
and much better understood than general blocks. This\nresult is obtained b
y using the theory of types and an isomorphism of\nHecke algebras. This is
a joint work with Jeffrey Adler\, Jessica Fintzen\,\nand Manish Mishra.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Heuer (University of Frankfurt)
DTSTART:20241127T160000Z
DTEND:20241127T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/144
DESCRIPTION:Title: Pro-étale vector bundles and the p-adic Simpson correspondence\
nby Ben Heuer (University of Frankfurt) as part of London number theory se
minar\n\nLecture held in Huxley 140\, Imperial College.\n\nAbstract\nI wil
l first explain how various classical problems in p-adic number theory suc
h as Sen theory can be reinterpreted geometrically in terms of vector bund
les on Scholze's pro-étale site. I will then explain how such pro-étale
vector bundles can be understood systematically by means of "p-adic non-ab
elian Hodge theory". This is closely related to Faltings' p-adic Simpson c
orrespondence\, relating p-adic representations of fundamental groups of p
-adic varieties to Higgs bundles. Finally\, I will sketch how moduli space
s of pro-étale vector bundles can help understand open questions in Sen t
heory and the p-adic Simpson correspondence.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siqi Yang (Imperial)
DTSTART:20241120T160000Z
DTEND:20241120T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/145
DESCRIPTION:Title: Hilbert modular forms and geometric modularity in quadratic case
\nby Siqi Yang (Imperial) as part of London number theory seminar\n\nLectu
re held in Huxley 140\, Imperial College.\n\nAbstract\nLet \\rho: G_\\Q \\
rightarrow \\GL_2(\\Fpbar) be a continuous\, odd\, irreducible representat
ion. The weight part of Serre's conjecture predicts the minimal weight k (
\\geq 2) such that \\rho arises from a modular eigenform of weight k. It i
s refined by Edixhoven to include the weight one forms by viewing mod p mo
dular forms as sections of certain line bundles on the special fibre of a
modular curve. One of the directions to generalise the weight part of Serr
e's conjecture is replacing Q with a totally real field F and replacing mo
dular forms with Hilbert modular forms. A conjecture in this setting is fo
rmulated by Buzzard\, Diamond and Jarvis\, where we have the notion of alg
ebraic modularity. On the other hand\, a generalisation of Edixhoven's ref
inement is considered by Diamond and Sasaki\, where we have the notion of
geometric modularity. I will discuss the relation between algebraic and ge
ometric modularity and show their consistency for the weights in a certain
cone\, under the assumption that F is a real quadratic field in which p i
s unramified.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Margherita Pagano (Imperial)
DTSTART:20241204T160000Z
DTEND:20241204T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/146
DESCRIPTION:Title: The wild Brauer-Manin obstruction\nby Margherita Pagano (Imperia
l) as part of London number theory seminar\n\nLecture held in Huxley 140\,
Imperial College.\n\nAbstract\nA way to study rational points on a variet
y is by looking at their image in the p-adic points. Some natural question
s that arise are the following: is there any obstruction to weak approxima
tion on the variety? Which primes might be involved in it? I will explain
how primes of good reduction can play a role in the Brauer-Manin obstructi
on to weak approximation\, with particular emphasis on the case of K3 surf
aces.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Deding Yang (Peking University)
DTSTART:20241211T160000Z
DTEND:20241211T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/147
DESCRIPTION:Title: Geometry of Shimura varieties and positivity of automorphic line bun
dles on Unitary Shimura varieties\nby Deding Yang (Peking University)
as part of London number theory seminar\n\nLecture held in Huxley 140\, Im
perial College.\n\nAbstract\nThe study of coherent cohomology on (the spec
ial fiber of) Shimura varieties has various applications to arithmetic pro
blems\, like\, \ncongruences of automorphic forms\, weight part of Serre's
conjecture\, liftability of mod p automorphic forms. One of the basic pro
blems is to prove certain automorphic line \nbundles are ample\, which yie
lds vanishing of coherent cohomology. In this talk\, we start from the amp
leness criterion of modular line bundles on Hilbert modular varieties\, \n
and then explain how to formulate a generalization to unitary Shimura vari
eties. Our final result works for split unitary Shimura varieties.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elvira Lupoian (University College London)
DTSTART:20250122T160000Z
DTEND:20250122T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/148
DESCRIPTION:Title: Ceresa cycles of modular curves\nby Elvira Lupoian (University C
ollege London) as part of London number theory seminar\n\nLecture held in
Room 505\, Department of Mathematics (25 Gordon St)\, University College L
ondon.\n\nAbstract\nThe Ceresa cycle is an algebraic cycle attached to smo
oth algebraic curve with a marked point\, which is always homologically tr
ivial. Ceresa proved that for a very general complex curve of genus at lea
st 3\, this cycle is not trivial as an element of the Chow group. Notably
\, hyperelliptic curves with a Weierstrass point have trivial Ceresa cycle
. There are few other explicit examples where triviality/non-triviality is
known. In this talk I will discuss the non-vanishing of the Ceresa cycle
attached to the modular curve $X_0(N)$.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Dotto (King's College London)
DTSTART:20250129T160000Z
DTEND:20250129T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/149
DESCRIPTION:Title: Gelfand--Kirillov dimension and mod $p$ cohomology for quaternion al
gebras\nby Andrea Dotto (King's College London) as part of London numb
er theory seminar\n\nLecture held in Room 505\, Department of Mathematics
(25 Gordon St)\, University College London.\n\nAbstract\nThe Gelfand--Kiri
llov dimension is a classical invariant which measures the size of smooth
representations of p-adic groups. It acquired particular relevance in the
mod $p$ Langlands program because of work of Breuil--Herzig--Hu--Morra--Sc
hraen\, who computed it for the mod $p$ cohomology of $\\mathrm{GL}_2$ ove
r totally real fields\, and used it to prove several structural properties
of the cohomology. In this talk we will present a simplified proof of thi
s result\, which has the added benefit of working unchanged for nonsplit i
nner forms of $\\mathrm{GL}_2$. This is joint work with Bao V. Le Hung.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Radu Toma (IMJ-PRG)
DTSTART:20250212T160000Z
DTEND:20250212T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/150
DESCRIPTION:Title: The size of newforms\nby Radu Toma (IMJ-PRG) as part of London n
umber theory seminar\n\nLecture held in Room 505\, Department of Mathemati
cs (25 Gordon St)\, University College London.\n\nAbstract\nGiven an $L^2$
-normalised newform for $\\mathrm{SL}(n)$\, how large are its values in te
rms of its level? The theory of quantum chaos suggests such a newform shou
ld be small. I will give an overview of how to show interesting upper boun
ds using spectral analysis\, Hecke operators\, and geometry of numbers. I
will present the first "non-trivial" bounds in higher rank and talk about
intermediate results of perhaps independent interest\, such as Atkin--Lehn
er operators for $\\mathrm{SL}(n)$ and a reduction theory with level struc
ture.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Han-Ung Kufner (Universität Regensburg)
DTSTART:20250219T160000Z
DTEND:20250219T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/151
DESCRIPTION:Title: Deligne's conjecture on the critical values of Hecke L-functions
\nby Han-Ung Kufner (Universität Regensburg) as part of London number the
ory seminar\n\nLecture held in Room 505\, Department of Mathematics (25 Go
rdon St)\, University College London.\n\nAbstract\nWe give a proof of Deli
gne's conjecture for critical algebraic Hecke characters\, which relates t
he special value of a Hecke L-function up to a rational factor with a cert
ain period. This generalises a result of Blasius in the case where the Hec
ke character is defined over a CM-field. In our approach\, we make use of
the recently constructed Eisenstein--Kronecker classes of Kings--Sprang\,
which allow for a cohomological interpretation of the L-value when the fie
ld of definition is an arbitrary totally imaginary number field. The key i
nsight is that these classes can be naturally regarded as de Rham classes
of Blasius' reflex motive\, which already played a key role in Blasius' pr
oof.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Keyes (King's College London)
DTSTART:20250305T160000Z
DTEND:20250305T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/152
DESCRIPTION:Title: Towards Artin's conjecture on $p$-adic quintic forms\nby Chris K
eyes (King's College London) as part of London number theory seminar\n\nLe
cture held in Room 505\, Department of Mathematics (25 Gordon St)\, Univer
sity College London.\n\nAbstract\nLet $K/\\mathbb{Q}_p$ be a finite extens
ion with residue field $\\mathbb{F}_q$ and suppose $f(x_0\, \\ldots\, x_n)
$ is a homogeneous polynomial of degree $d$ over $K$. A conjecture\, origi
nally due to Artin\, states that when $d$ is prime and $n \\geq d^2$\, $f=
0$ has a nontrivial solution in $K$. This conjecture is known in degrees 2
and 3 due to Hasse and Lewis\, respectively. It is also "asymptotically t
rue\," due to work of Ax and Kochen\, in that it holds when $q$ is suffici
ently large with respect to $d$\, though this is difficult to make effecti
ve. In this talk\, we present recent joint work with Lea Beneish in which
we prove the quintic version of the conjecture holds if $q \\geq 7$. Our m
ethods include both a refinement to a geometric approach of Leep and Yeoma
ns (who showed $q \\geq 47$ suffices) and a significant computational comp
onent.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joni Teräväinen (University of Cambridge)
DTSTART:20250319T160000Z
DTEND:20250319T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/153
DESCRIPTION:Title: On the exceptional set in the $abc$ conjecture\nby Joni Terävä
inen (University of Cambridge) as part of London number theory seminar\n\n
Lecture held in Room 505\, Department of Mathematics (25 Gordon St)\, Univ
ersity College London.\n\nAbstract\nThe well-known $abc$ conjecture assert
s that for any coprime triple of positive integers satisfying $a+b=c$ the
squarefree radical of $abc$ satisfies a certain strong inequality. In this
talk\, I will discuss a proof giving the first power-saving improvement o
ver the trivial bound for the number of exceptions to this conjecture. The
proof is based on a combination of various methods for counting rational
points on curves\, and a combinatorial analysis to patch these cases toget
her. This is joint work with Tim Browning and Jared Lichtman.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harry Schmidt (University of Warwick)
DTSTART:20250326T160000Z
DTEND:20250326T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/154
DESCRIPTION:Title: Effective estimates for André-Oort in families of elliptic curves\nby Harry Schmidt (University of Warwick) as part of London number theo
ry seminar\n\nLecture held in Room 505\, Department of Mathematics (25 Gor
don St)\, University College London.\n\nAbstract\nI will present joint wor
k with Binyamini\, Jones\, Thomas in which we manage to give uniform and e
ffective proofs of the André-Oort conjecture for families of elliptic cur
ves. Time permitting\, I will discuss possible generalizations of our resu
lt to families of abelian varieties.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Herbert Gangl (Durham University)
DTSTART:20250312T160000Z
DTEND:20250312T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/155
DESCRIPTION:Title: Multiple polylogarithms\, and Zagier's Conjecture revisited\nby
Herbert Gangl (Durham University) as part of London number theory seminar\
n\nLecture held in Room 505\, Department of Mathematics (25 Gordon St)\, U
niversity College London.\n\nAbstract\nDirichlet related the residue at $s
=1$ of the Dedekind zeta function of a number field $F$ (a slight generali
sation of the famous Riemann zeta function) to two important arithmetical
notions: the size of the ideal class group and the `volume' of the unit gr
oup in the number ring $\\mathcal{O}_F$ of $F$. Generalising this surprisi
ng connection\, the special values of the Dedekind zeta function of a numb
er field $F$ at integer argument $n$ should\, according to Zagier's Polylo
garithm Conjecture\, be expressed via a determinant of $F$-values of the $
n$-th polylogarithm function. Goncharov laid out a vast program incorporat
ing this conjecture using properties of multiple polylogarithms and the st
ructure of a motivic Lie coalgebra.\nIn this impressionist talk I intend t
o give a rough idea of the developments from the early days on\, avoiding
most of the technical bits\, and also hint at a number of recent results f
or higher weight\, some in joint work with\, or developed by\, S. Charlto
n\, D. Radchenko as well as D. Rudenko and his collaborators.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Graham (University of Oxford)
DTSTART:20250205T160000Z
DTEND:20250205T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/156
DESCRIPTION:Title: The exceptional zero conjecture for $\\mathrm{GL}(3)$\nby Andrew
Graham (University of Oxford) as part of London number theory seminar\n\n
Lecture held in Room 505\, Department of Mathematics (25 Gordon St)\, Univ
ersity College London.\n\nAbstract\nIf $E$ is an elliptic curve over $\\ma
thbb{Q}$ with split multiplicative reduction at $p$\, then the $p$-adic $L
$-function associated with $E$ vanishes at $s=1$ independently of whether
the complex $L$-function vanishes. In this case\, one has an "exceptional
zero formula" relating the first derivative of the $p$-adic $L$-function t
o the complex $L$-function multiplied by a certain L-invariant. This L-inv
ariant can be interpreted in several ways -- on the automorphic side for e
xample\, L-invariants parameterise part of the $p$-adic local Langlands co
rrespondence for $\\mathrm{GL}_2(\\mathbb{Q}_p)$.\n\nIn this talk\, I will
discuss an exceptional zero formula for (not necessarily essentially self
-dual) regular algebraic\, cuspidal automorphic representations of $\\math
rm{GL}_3$ which are Steinberg at $p$. The formula involves an automorphic
L-invariant constructed by Gehrmann. Joint work with Daniel Barrera and Ch
ris Williams.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandros Konstantinou (University College London)
DTSTART:20250115T160000Z
DTEND:20250115T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/157
DESCRIPTION:Title: Isogeny decompositions and the Tate--Shafarevich group modulo square
s\nby Alexandros Konstantinou (University College London) as part of L
ondon number theory seminar\n\nLecture held in Room 505\, Department of Ma
thematics (25 Gordon St)\, University College London.\n\nAbstract\nWe pres
ent a method for decomposing abelian varieties up to isogeny using group a
ctions and finite group representation theory. As an application\, we show
that given a square-free natural number n\, there exists an abelian varie
ty with finite Tate--Shafarevich group of order n times a square.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samir Siksek (University of Warwick)
DTSTART:20250226T143000Z
DTEND:20250226T153000Z
DTSTAMP:20260404T111112Z
UID:LNTS/158
DESCRIPTION:Title: Galois groups of low degree points on curves\nby Samir Siksek (U
niversity of Warwick) as part of London number theory seminar\n\nLecture h
eld in Room 505\, Department of Mathematics (25 Gordon St)\, University Co
llege London.\n\nAbstract\nLow degree points on curves have been subject o
f intense\nstudy for several decades\, but little attention has been paid
to the\nGalois groups of those points. In this talk we recall primitive gr
oup\nactions\, and focus on low degree points whose Galois group is\nprimi
tive. We shall see that such points are relatively rare\, and that\nthey i
nterfere with each other. This talk is based on joint work with\nMaleeha K
hawaja.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeehoon Park (Seoul National University)
DTSTART:20250205T130000Z
DTEND:20250205T140000Z
DTSTAMP:20260404T111112Z
UID:LNTS/159
DESCRIPTION:Title: BF path integrals for elliptic curves and $p$-adic L-functions\n
by Jeehoon Park (Seoul National University) as part of London number theor
y seminar\n\nLecture held in Room 505\, Department of Mathematics (25 Gord
on St)\, University College London.\n\nAbstract\nIn this talk\, I will exp
lain an arithmetic path integral formula for the inverse $p$-adic absolute
values of the $p$-adic L-functions of elliptic curves over the rational n
umbers with good ordinary reduction at an odd prime $p$ based on the Iwasa
wa main conjecture and Mazur’s control theorem. The talk is based on a j
oint work with Junyeong Park.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dario Beraldo (UCL)
DTSTART:20250226T160000Z
DTEND:20250226T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/160
DESCRIPTION:Title: The Deligne--Milnor formula\nby Dario Beraldo (UCL) as part of L
ondon number theory seminar\n\nLecture held in Room 505\, Department of Ma
thematics (25 Gordon St)\, University College London.\n\nAbstract\nLet $X
\\to S$ be a family of algebraic varieties parametrized by an infinitesima
l disk $S$\, possibly of mixed characteristic. Bloch's conductor conjectur
e expresses the difference of the Euler characteristics of the special and
generic fibers in algebraic and arithmetic terms. I'll describe a proof o
f some new cases of this conjecture\, including the case of isolated singu
larities. The latter was a conjecture of Deligne generalizing Milnor's for
mula on vanishing cycles. (This is based on joint work with Massimo Pippi.
)\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilaria Viglino (EPFL)
DTSTART:20250430T150000Z
DTEND:20250430T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/161
DESCRIPTION:Title: Moment estimates of module lattice points for effective lattice cons
tructions\nby Ilaria Viglino (EPFL) as part of London number theory se
minar\n\nLecture held in K2.40\, King's Building\, King's College London\,
WC2R 2LS.\n\nAbstract\nWe examine the moments of the number of lattice po
ints in a fixed ball of volume $V$ for lattices in Euclidean space which a
re modules over the ring of integers of a number field $K$. In particular\
, we show that moments obtained for “lifts of codes” to $\\mathcal{O}_
K$-modules converge to the Rogers integral formula for the space of free $
\\mathcal{O}_K$-module lattices. This extends work of Rogers for $\\mathbb
{Z}$-lattices. Joint work with Maryna Viazovska\, Nihar Gargava and Vlad S
erban.\n\nVisitors will need to sign in at the Strand Building reception d
esk and receive an events sticker to access the building.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Santens (University of Cambridge)
DTSTART:20250507T150000Z
DTEND:20250507T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/162
DESCRIPTION:Title: Leading constant in Malle's conjecture\nby Tim Santens (Universi
ty of Cambridge) as part of London number theory seminar\n\nLecture held i
n K2.40\, King's Building\, King's College London\, WC2R 2LS.\n\nAbstract\
nLet G be a finite permutation group\, Malle has put forward a conjecture
on the number of G-extensions of a number field of bounded discriminant. T
here exists a superficially similar conjecture by Manin on the number of p
oints of bounded height on varieties. In this talk I will discuss recent e
fforts to interpret Malle's conjecture as a form of Manin's conjecture for
the stack BG. Based on this analogy me and Loughran have given a conjectu
ral interpretation of the leading constant in Malle's conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesca Balestrieri (American University of Paris)
DTSTART:20250514T150000Z
DTEND:20250514T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/163
DESCRIPTION:Title: The quadratic Manin-Peyre conjecture for del Pezzo surfaces\nby
Francesca Balestrieri (American University of Paris) as part of London nu
mber theory seminar\n\nLecture held in K2.40\, King's Building\, King's Co
llege London\, WC2R 2LS.\n\nAbstract\nIn this talk\, we outline a general
framework for the study of the "quadratic" Manin-Peyre conjecture (i.e. th
e Manin-Peyre conjecture for symmetric squares of varieties) for del Pezzo
surfaces. We then apply this framework (in conjunction with\, among other
things\, some novel lattice counting techniques) to prove that the quadra
tic Manin-Peyre conjecture holds for an infinite family of non-split quadr
ics. This is joint work with Kevin Destagnol\, Julian Lyczak\, Jennifer Pa
rk\, and Nick Rome.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katerina Santicola (University of Warwick)
DTSTART:20250521T150000Z
DTEND:20250521T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/164
DESCRIPTION:Title: Sharp interpolation of rational points and post-quantum cryptography
\nby Katerina Santicola (University of Warwick) as part of London numb
er theory seminar\n\nLecture held in K2.40\, King's Building\, King's Coll
ege London\, WC2R 2LS.\n\nAbstract\nFor curves\, it is more or less conjec
tured that the Chabauty method with the Mordell–Weil sieve will give a p
olynomial-time algorithm for finding the sets of rational points over numb
er fields. Whether this is true for arbitrary varieties is unclear. Over f
inite fields the situation is different. This is the MQ problem\, which is
NP-complete. This problem is the essence of multivariate cryptography and
forms the basis of most post-quantum signature schemes. The aim of this t
alk is to give an overview of the interpolation of rational subvarieties a
nd discuss where this fits in with modern cryptography.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART:20250604T150000Z
DTEND:20250604T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/165
DESCRIPTION:Title: Smooth discrepancy and Littlewood’s conjecture\nby Sam Chow (U
niversity of Warwick) as part of London number theory seminar\n\nLecture h
eld in K2.40\, King's Building\, King's College London\, WC2R 2LS.\n\nAbst
ract\nWe establish a deterministic analogue of Beck’s local-to-global pr
inciple for Kronecker sequences. This gives rise to a novel reformulation
of Littlewood’s conjecture in Diophantine approximation.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Bloom (University of Manchester)
DTSTART:20250611T150000Z
DTEND:20250611T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/166
DESCRIPTION:Title: Numbers with small digits in multiple bases\nby Thomas Bloom (Un
iversity of Manchester) as part of London number theory seminar\n\nLecture
held in K2.40\, King's Building\, King's College London\, WC2R 2LS.\n\nAb
stract\nAn old conjecture of Graham asks whether there are infinitely many
integers $n$ such that $\\binom{2n}{n}$ is coprime to 105. This is equiva
lent to asking whether there are infinitely many integers which only have
the digits 0\,1 in base 3\, 0\,1\,2 in base 5\, and 0\,1\,2\,3 in base 7.
In general\, one can ask whether there are infinitely many integers which
only have 'small' digits in multiple bases simultaneously. For two bases t
his was established in 1975 by Erdos\, Graham\, Ruzsa\, and Straus\, but t
he case of three or more bases is much more mysterious. I will discuss rec
ent joint work with Ernie Croot\, in which we prove that (assuming the ba
ses are sufficiently large) there are infinitely many integers such that a
lmost all of the digits are small in all bases simultaneously.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pär Kurlberg (KTH)
DTSTART:20250618T150000Z
DTEND:20250618T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/167
DESCRIPTION:Title: A Poincare section for horocycle flows: escape of mass\nby Pär
Kurlberg (KTH) as part of London number theory seminar\n\nLecture held in
K2.40\, King's Building\, King's College London\, WC2R 2LS.\n\nAbstract\nM
otivated by a hyperbolic analog of the Lester-Wigman\n"vanishing area corr
elations"-conjecture for euclidean lattice points we\ninvestigate the dyna
mical properties of a natural choice of a Poincare\nsection\, associated w
ith H/SL(2\,Z)\, and the horocycle flow on the upper\nhalf plane H. Since
the horocycle *flow* is mixing\, one might hope for\nan easy proof of vani
shing area correlations by showing that the\nPoincare map is mixing. Howev
er\, not only is the Poincare map\nnon-mixing\; even equidistribution/ergo
dicity breaks down badly due to\nescape of mass. Amusingly\, we can still
show vanishing of area\ncorrelations (but "for the wrong reason".)\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/167/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ned Carmichael (King's College London)
DTSTART:20250625T150000Z
DTEND:20250625T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/168
DESCRIPTION:Title: Sums of Hecke Eigenvalues\nby Ned Carmichael (King's College Lon
don) as part of London number theory seminar\n\nLecture held in K2.40\, Ki
ng's Building\, King's College London\, WC2R 2LS.\n\nAbstract\nMotivated b
y the Dirichlet divisor problem\, which asks for the best possible error t
erm in the classical asymptotic formula for sums of the divisor function\,
we consider sums of Hecke eigenvalues attached to holomorphic cusp forms.
We discuss some recent results\, which reveal interesting transitions in
the average size of these sums as the length of the sums varies relative t
o the weight of the forms.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Happy Uppal (University of Bristol)
DTSTART:20250528T150000Z
DTEND:20250528T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/169
DESCRIPTION:Title: Lines on del Pezzo surfaces\nby Happy Uppal (University of Brist
ol) as part of London number theory seminar\n\nLecture held in K2.40\, Kin
g's Building\, King's College London\, WC2R 2LS.\n\nAbstract\nOne of the c
rowning achievements of classical algebraic geometry is the Cayley--Salmon
theorem\, which states that any smooth cubic surface over an algebraicall
y closed field contains exactly 27 lines. Over more general fields\, howev
er\, the situation becomes more subtle: the number of lines depends on the
arithmetic of the field. Segre classified the possible numbers of lines t
hat can appear on a cubic surface over arbitrary fields and showed that al
l such line counts can be realised over the rational numbers.\n\nIn this t
alk\, I will discuss joint work with Enis Kaya\, Stephen McKean\, and Sam
Streeter\, in which we extend this perspective to del Pezzo surfaces---a c
lass of surfaces that includes cubic surfaces. We investigate which line c
ounts can occur on del Pezzo surfaces over general fields and how these co
unts are influenced by the arithmetic of the field. We also explore the an
alogous question for conic bundles.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Morgan (Cambridge University)
DTSTART:20251008T150000Z
DTEND:20251008T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/170
DESCRIPTION:Title: On the Hasse principle in twist families\nby Adam Morgan (Cambri
dge University) as part of London number theory seminar\n\nLecture held in
Imperial College London\, Room 139.\n\nAbstract\nI will discuss joint wor
k with Alex Bartel in which we study the frequency of failures of the Hass
e principle in quadratic twist families of torsors under an abelian variet
y. The main technical ingredient is a result on the distribution of 2-Selm
er groups in families of twists defined by Frobenian conditions.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chengyang Bao (Imperial College London)
DTSTART:20251015T150000Z
DTEND:20251015T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/171
DESCRIPTION:Title: Applications of patching the coherent cohomology of modular curves\nby Chengyang Bao (Imperial College London) as part of London number th
eory seminar\n\nLecture held in Imperial College London\, Room 139.\n\nAbs
tract\nWe apply the Taylor--Wiles--Kisin patching method to study certain
partial normalizations of crystalline deformation rings associated with tw
o-dimensional representations \\bar{r} : G_{\\Q_p} \\to \\GL_2(\\F)\, wher
e $\\F$ is a finite field of characteristic $p \\ge 5$. Using the $q$-expa
nsion principle\, we obtain a multiplicity-one result\, which implies that
the partial normalization of the crystalline deformation ring is Cohen--M
acaulay. As applications\, we give a simple criterion for when a crystalli
ne deformation ring coincides with its partial normalization\, thereby est
ablishing new cases where these rings are Cohen--Macaulay. We also prove a
Zariski-density result for crystalline points in characteristic $p$\, and
we apply our method to deduce a multiplicity-one result for Serre's mod-$
p$ quaternionic modular forms. \n \nMost of these results originated from
attempts to explain computational data from my thesis on computing crystal
line deformation rings via the Taylor--Wiles--Kisin patching method. I wil
l conclude with some expected properties of crystalline deformation rings
suggested by the data that remain open.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Scavia (Université Sorbonne Paris Nord)
DTSTART:20251022T150000Z
DTEND:20251022T160000Z
DTSTAMP:20260404T111112Z
UID:LNTS/172
DESCRIPTION:Title: The lifting problem for Galois representations\nby Federico Scav
ia (Université Sorbonne Paris Nord) as part of London number theory semin
ar\n\nLecture held in Imperial College London\, Room 139.\n\nAbstract\nFor
every finite group H and every finite H-module A\, we determine the subgr
oup of negligible classes in H^2(H\, A)\, in the sense of Serre\, over fie
lds with enough roots of unity. As a consequence\, we show that for every
odd prime p and every field F containing a primitive p-th root of unity\,
there exists a continuous 3-dimensional mod p representation of the absolu
te Galois group of F(x_1\, ...\, x_p) which does not lift modulo p^2. We a
lso construct continuous 5-dimensional Galois representations mod 2 which
do not lift modulo 4. This answers a question of Khare and Serre\, and dis
proves a conjecture of Florence. This is joint work with Alexander Merkurj
ev.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentijn Karemaker (UvA University of Amsterdam)
DTSTART:20251029T160000Z
DTEND:20251029T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/173
DESCRIPTION:Title: Arithmetic invariants of supersingular abelian varieties\nby Val
entijn Karemaker (UvA University of Amsterdam) as part of London number th
eory seminar\n\nLecture held in Imperial College London\, Room 139.\n\nAbs
tract\nWe will study the moduli space of abelian varieties in characterist
ic p and in particular its supersingular locus S_g. We will discuss when t
his locus is geometrically irreducible\, thereby solving a “class numbe
r one problem” or “Gauss problem” for the number of irreducible co
mponents\; and when a polarised abelian variety is determined by its p-di
visible group\, solving a Gauss problem for central leaves\, which are th
e loci consisting of points whose associated p-divisible groups are isomor
phic. Furthermore\, Oort conjectured that all generic points of S_g have
automorphism group {+/- 1}. We will present our results that settle Oort
’s conjecture for g=2\,3\,4\, and for all higher even dimensions when p
>= 5. This is based on joint works with Ibukiyama and Yu.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/173/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Bartlett (Queen Mary University of London)
DTSTART:20251126T160000Z
DTEND:20251126T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/174
DESCRIPTION:Title: Isolating the Hodge type in moduli spaces of crystalline Galois repr
esentations\nby Robin Bartlett (Queen Mary University of London) as pa
rt of London number theory seminar\n\nLecture held in Imperial College Lon
don\, Room 139.\n\nAbstract\nModuli spaces of representations of the absol
ute Galois group of a p-adic field play an important role in various aspec
ts of the Langlands correspondence. In this talk I will focus on cases in
which the coefficients of these representations also have characteristic p
\, and discuss joint work with Bao Le-Hung and Brandon Levin in which we c
ontrol the singularities of these moduli spaces in several new cases. One
new ingredient is a description of integral conditions\, derived from Plü
cker coordinates on the affine Grassmannian\, which cut out the locus with
a specific Hodge type. This works for any ramification degree\, and as an
application we can extend modularity lifting theorems proved by Kisin for
two dimensional Galois representations of a totally real number field\, t
o three dimensional representations.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Groen (UvA University of Amsterdam)
DTSTART:20251203T160000Z
DTEND:20251203T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/175
DESCRIPTION:Title: Ekedahl-Oort strata of double covers in characteristic 2.\nby St
even Groen (UvA University of Amsterdam) as part of London number theory s
eminar\n\nLecture held in Imperial College London\, Room 139.\n\nAbstract\
nThis talk concerns a variant of the Schottky problem\, which asks to clas
sify Jacobians among all abelian varieties. In characteristic p\, there is
a rich extra structure to consider. Namely\, in characteristic p\, abelia
n varieties can be partitioned into so-called Ekedahl-Oort strata\, within
which all abelian varieties have isomorphic p-torsion group schemes. From
this point of view\, it is fruitful to investigate which p-torsion group
schemes can occur as the p-torsion of the Jacobian of a (specified type of
) curve. In this talk\, we treat the 2-torsion of curves in characteristic
2 that admit a separable double cover to another curve. Through an analys
is of the first De Rham cohomology\, we prove that the p-torsion of a doub
le cover of an ordinary curve is determined by the ramification breaks of
the cover. This generalizes a result by Elkin and Pries\, where the base c
urve is the projective line and the covers are hyperelliptic curves. When
the base curve is not ordinary\, we establish bounds on the Ekedahl-Oort t
ype of the cover.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/175/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Domenico Valloni (EPFL Lausanne)
DTSTART:20251210T160000Z
DTEND:20251210T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/176
DESCRIPTION:Title: p-torsion Brauer classes in positive characteristic\nby Domenico
Valloni (EPFL Lausanne) as part of London number theory seminar\n\nLectur
e held in Imperial College London\, Room 139.\n\nAbstract\nIn this talk\,
we will study p-torsion Brauer classes arising from differential forms in
characteristic p. We will then explain how such classes contribute to the
Brauer–Manin obstruction. As an application\, we determine the Brauer–
Manin set of varieties that possess “many differential forms\,” and we
obtain new results on the Brauer–Manin set of supersingular K3 surfaces
.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/176/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Livia Grammatica (Strasbourg University)
DTSTART:20251105T160000Z
DTEND:20251105T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/177
DESCRIPTION:Title: Smoothness of the formal Brauer group\nby Livia Grammatica (Stra
sbourg University) as part of London number theory seminar\n\nLecture held
in Imperial College London\, Room 139.\n\nAbstract\nLet X be a smooth and
proper variety over an algebraically closed field of characteristic p. Th
e formal Brauer group of X is the functor which parametrizes deformations
of the trivial Brauer class of X. Under mild assumptions\, it is represent
able by a formal group\, closely related to the p-torsion of Br(X). We wil
l give criteria for this formal group to be smooth in terms of the crystal
line cohomology of X\, thus providing a partial answer to a question of Ar
tin-Mazur. The strategy is to relate the formal Brauer group to crystallin
e cohomology using the relationship between fppf cohomology\, crystalline
cohomology and the Nygaard filtration. These criteria can be used in pract
ice to produce varieties with non-smooth formal Brauer group\, which are c
onstructed as higher-dimensional analogues of Igusa's surface with non-smo
oth Picard group.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/177/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Gouthier (MPIM Bonn)
DTSTART:20251112T160000Z
DTEND:20251112T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/178
DESCRIPTION:Title: Infinitesimal rational actions on curves\nby Bianca Gouthier (MP
IM Bonn) as part of London number theory seminar\n\nLecture held in Imperi
al College London\, Room 139.\n\nAbstract\nFor any finite $k$-group scheme
$G$ acting rationally on a $k$-variety $X$\, if the action is generically
free then the dimension of $Lie (G)$ is upper bounded by the dimension of
the variety.\nThis inequality turns out to be also a sufficient condition
for the existence of such actions\, when $k$ is a perfect field of positi
ve characteristic and $G$ is infinitesimal commutative trigonalizable.\nIn
this talk\, we will specialize to the case in which $X$ is a curve. First
\, we will give an explicit description of all the infinitesimal commutati
ve unipotent group schemes $G$ with a generically free rational action on
$X$ when $k$ is algebraically closed. We will then see how these actions c
an be constructed\, focusing on the case in which $G$ is the $p$-torsion o
f a supersingular elliptic curve.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/178/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sudip Pandit (Kings College London)
DTSTART:20251119T160000Z
DTEND:20251119T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/179
DESCRIPTION:Title: Delta isocrystal and crystalline cohomology of abelian schemes\n
by Sudip Pandit (Kings College London) as part of London number theory sem
inar\n\nLecture held in Imperial College London\, Room 139.\n\nAbstract\nG
iven an abelian scheme A over a p-adic ring\, using the theory of arithmet
ic jets of A\, one can associate a filtered F-isocrystal\, which is referr
ed to as the delta isocrystal associated with A. The delta isocrystal admi
ts a natural map to the Hodge sequence of the first de Rham cohomology of
A. Recently\, we have shown that the Frobenius operator on the delta isocr
ystal is compatible with the crystalline Frobenius operator on the de Rham
cohomology (under the de Rham-crystalline comparison isomorphism). This a
llows us to derive a comparison result between the delta isocrystal and th
e crystalline cohomology of abelian schemes in the category of filtered F-
isocrystals. This talk is partly based on joint work with Lance Gurney and
Arnab Saha.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/179/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephanie Chan (University College London)
DTSTART:20260121T160000Z
DTEND:20260121T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/180
DESCRIPTION:Title: Pointwise bounds for 3-torsion\nby Stephanie Chan (University Co
llege London) as part of London number theory seminar\n\nLecture held in R
oom 505\, UCL Maths building\, 25 Gordon Street.\n\nAbstract\nFor $\\ell$
an odd prime number and $d$ a squarefree integer\, a notable problem in ar
ithmetic statistics is to give pointwise bounds for the size of the $\\ell
$-torsion of the class group of $\\mathbb{Q}(\\sqrt{d})$. This is in gener
al a difficult problem\, and unconditional pointwise bounds are only avail
able for $\\ell = 3$ due to work of Pierce\, Helfgott–Venkatesh and Elle
nberg–Venkatesh. The current record due to Ellenberg–Venkatesh is $h_3
(d) \\ll_\\epsilon d^{1/3 + \\epsilon}$. We will discuss how to improve th
is to $h_3(d) \\ll d^{0.32}$. This is joint work with Peter Koymans.\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/180/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga
DTSTART:20260128T160000Z
DTEND:20260128T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/181
DESCRIPTION:by Jef Laga as part of London number theory seminar\n\nLecture
held in Room 505\, UCL Maths building\, 25 Gordon Street.\nAbstract: TBA\
n
LOCATION:https://stable.researchseminars.org/talk/LNTS/181/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maleeha Khawaja
DTSTART:20260204T160000Z
DTEND:20260204T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/182
DESCRIPTION:by Maleeha Khawaja as part of London number theory seminar\n\n
Lecture held in Room 505\, UCL Maths building\, 25 Gordon Street.\nAbstrac
t: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/182/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Le Fourn
DTSTART:20260211T160000Z
DTEND:20260211T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/183
DESCRIPTION:by Samuel Le Fourn as part of London number theory seminar\n\n
Lecture held in Room 505\, UCL Maths building\, 25 Gordon Street.\nAbstrac
t: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/183/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trajan Hammonds
DTSTART:20260218T160000Z
DTEND:20260218T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/184
DESCRIPTION:by Trajan Hammonds as part of London number theory seminar\n\n
Lecture held in Room 505\, UCL Maths building\, 25 Gordon Street.\nAbstrac
t: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/184/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Bartel
DTSTART:20260225T160000Z
DTEND:20260225T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/185
DESCRIPTION:by Alex Bartel as part of London number theory seminar\n\nLect
ure held in Room 505\, UCL Maths building\, 25 Gordon Street.\nAbstract: T
BA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/185/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Izzy Rendell (KCL)
DTSTART:20260304T160000Z
DTEND:20260304T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/186
DESCRIPTION:by Izzy Rendell (KCL) as part of London number theory seminar\
n\nLecture held in Room 505\, UCL Maths building\, 25 Gordon Street.\nAbst
ract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/186/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xinchen Miao (Mathematisches Institut Bonn)
DTSTART:20260318T143000Z
DTEND:20260318T153000Z
DTSTAMP:20260404T111112Z
UID:LNTS/187
DESCRIPTION:by Xinchen Miao (Mathematisches Institut Bonn) as part of Lond
on number theory seminar\n\nLecture held in Room 505\, UCL Maths building\
, 25 Gordon Street.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/187/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Rawson (University of Glasgow)
DTSTART:20260311T160000Z
DTEND:20260311T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/190
DESCRIPTION:by James Rawson (University of Glasgow) as part of London numb
er theory seminar\n\nLecture held in University College London\, Room 505.
\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/190/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sacha Mangerel (Durham University)
DTSTART:20260325T160000Z
DTEND:20260325T170000Z
DTSTAMP:20260404T111112Z
UID:LNTS/191
DESCRIPTION:by Sacha Mangerel (Durham University) as part of London number
theory seminar\n\nLecture held in University College London\, Room 505.\n
Abstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/LNTS/191/
END:VEVENT
END:VCALENDAR