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BEGIN:VEVENT
SUMMARY:Vesselin Dimitrov (University of Toronto)
DTSTART:20200317T203000Z
DTEND:20200317T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/1
DESCRIPTION:Title: An arithmetic holonomicity criterion and irrationality of the 2-adic
period $\\zeta_2(5)$\nby Vesselin Dimitrov (University of Toronto) as
part of MIT number theory seminar\n\n\nAbstract\nI will present a new arit
hmetic criterion for a formal power\nseries to satisfy a linear ODE on an
affine curve over a global field.\nThis result characterizes the holonomic
functions by a sharp positivity\ncondition on a suitably defined arithmet
ic degree for an adelic set where\na given formal power series is analytic
. As an application\, based on\nCalegari's method with overconvergent p-ad
ic modular forms\, we derive an\nirrationality proof of the Leopoldt-Kubot
a 2-adic zeta value $\\zeta_2(5)$.\nThis is a joint work in progress with
Frank Calegari and Yunqing Tang.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicole Looper (Brown University)
DTSTART:20200331T203000Z
DTEND:20200331T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/2
DESCRIPTION:Title: Equidistribution techniques in arithmetic dynamics\nby Nicole Loo
per (Brown University) as part of MIT number theory seminar\n\n\nAbstract\
nThis talk is about the arithmetic of points of small canonical height\nre
lative to dynamical systems over number fields\, particularly those\naspec
ts amenable to the use of equidistribution techniques. Past milestones\nin
the subject include the proof of the Manin-Mumford Conjecture given by\nS
zpiro-Ullmo-Zhang\, and Baker-DeMarco's work on the finiteness of common\n
preperiodic points of rational functions. Recently\, quantitative\nequidis
tribution techniques have emerged both as a way of improving upon\nsome of
these old results\, and as an avenue to studying previously\ninaccessible
problems\, such as the Uniform Boundedness Conjecture of Morton\nand Silv
erman. I will describe the key ideas behind these developments\, and\nrais
e related questions for future research.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uriya First (University of Haifa)
DTSTART:20200428T203000Z
DTEND:20200428T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/3
DESCRIPTION:Title: Generation of algebras and versality of torsors\nby Uriya First (
University of Haifa) as part of MIT number theory seminar\n\n\nAbstract\nT
he primitive element theorem states that every finite separable field\next
ension L/K is generated by a single element. An almost equally known\nfolk
lore fact states that every central simple algebra over a field can be\nge
nerated by 2-elements.\n\nI will discuss two recent works with Zinovy Reic
hstein (one is forthcoming)\nwhere we establish global analogues of these
results. In more detail\, over\na ring R (or a scheme X)\, separable field
extensions and central simple\nalgebras globalize to finite etale algebra
s and Azumaya algebras\,\nrespectively. We show that if R is of finite typ
e over an infinite field K\nand has Krull dimension d\, then every finite
etale R-algebra is generated\nby d+1 elements and every Azumaya R-algebra
of degree n is generated by\n2+floor(d/[n-1]) elements. The case d=0 recov
ers the well-known facts\nstated above. Recent works of B. Williams\, A.K.
Shukla and M. Ojanguren\nshow that these bounds are tight in the etale ca
se and suggest that they\nshould also be tight in the Azumaya case.\n\nThe
proof makes use of principal homogeneous G-bundles T-->X (G is an\naffine
algebraic group over K) which can specialize to any principal\nhomogeneou
s G-bundle over an affine K-variety of dimension at most d. In\nparticular
\, such G-bundles exist for all G and d.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Vonk (Institute for Advanced Study)
DTSTART:20200505T203000Z
DTEND:20200505T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/4
DESCRIPTION:Title: Singular moduli for real quadratic fields\nby Jan Vonk (Institute
for Advanced Study) as part of MIT number theory seminar\n\n\nAbstract\nI
n the early 20th century\, Hecke studied the diagonal restrictions of Eise
nstein series over real quadratic fields. An infamous sign error caused hi
m to miss an important feature\, which later lead to highly influential de
velopments in the theory of complex multiplication (CM) initiated by Gross
and Zagier in their famous work on Heegner points on elliptic curves. In
this talk\, we will explore what happens when we replace the imaginary qua
dratic fields in CM theory with real quadratic fields\, and propose a fram
ework for a tentative 'RM theory'\, based on the notion of rigid meromorph
ic cocycles\, introduced in joint work with Henri Darmon. I will discuss s
everal of their arithmetic properties\, and their apparent relevance in th
e study of explicit class field theory of real quadratic fields\, the cons
truction of rational points on elliptic curves\, and the theory of Borcher
ds lifts. This concerns various joint works\, with Henri Darmon\, Alice Po
zzi\, and Yingkun Li.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jessica Fintzen (Cambridge/Duke/IAS)
DTSTART:20200908T143000Z
DTEND:20200908T153000Z
DTSTAMP:20260404T094147Z
UID:MITNT/6
DESCRIPTION:Title: Representations of p-adic groups and applications\nby Jessica Fin
tzen (Cambridge/Duke/IAS) as part of MIT number theory seminar\n\n\nAbstra
ct\nThe Langlands program is a far-reaching collection of conjectures that
relate different areas of mathematics including number theory and represe
ntation theory. A fundamental problem on the representation theory side of
the Langlands program is the construction of all (irreducible\, smooth\,
complex) representations of p-adic groups. \n\nI will provide an overview
of our understanding of the representations of p-adic groups\, with an emp
hasis on recent progress. \n\nI will also outline how new results about th
e representation theory of p-adic groups can be used to obtain congruences
between arbitrary automorphic forms and automorphic forms which are super
cuspidal at p\, which is joint work with Sug Woo Shin. This simplifies ear
lier constructions of attaching Galois representations to automorphic repr
esentations\, i.e. the global Langlands correspondence\, for general linea
r groups. Moreover\, our results apply to general p-adic groups and have t
herefore the potential to become widely applicable beyond the case of the
general linear group.\n\nNote the this talk will take place at 10:30 rathe
r than 16:30 (Eastern time).\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Lawrence (University of Chicago)
DTSTART:20200915T203000Z
DTEND:20200915T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/7
DESCRIPTION:Title: The Shafarevich conjecture for hypersurfaces in abelian varieties
\nby Brian Lawrence (University of Chicago) as part of MIT number theory s
eminar\n\n\nAbstract\nLet K be a number field\, S a finite set of primes o
f O_K\, and g a positive integer. Shafarevich conjectured\, and Faltings
proved\, that there are only finitely many curves of genus g\, defined ove
r K and having good reduction outside S. Analogous results have been prov
en for other families\, replacing "curves of genus g" with "K3 surfaces"\,
"del Pezzo surfaces" etc.\; these results are also called Shafarevich con
jectures. There are good reasons to expect the Shafarevich conjecture to
hold for many families of varieties: the moduli space should have only fin
itely many integral points.\n\nWill Sawin and I prove this for hypersurfac
es in abelian varieties of dimension not equal to 3.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shou-Wu Zhang (Princeton University)
DTSTART:20200922T203000Z
DTEND:20200922T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/8
DESCRIPTION:Title: Decomposition theorems for arithmetic cycles\nby Shou-Wu Zhang (P
rinceton University) as part of MIT number theory seminar\n\n\nAbstract\nW
e will describe some decomposition theorems for cycles over polarized
varieties in both local and global settings under some conjectures of
Lefschetz type. In local settings\, our decomposition theorems are esse
ntially non-archimedean analogues of ``harmonic forms" on Kahler manifol
ds. As an application\, we will define a notion of ``admissible pairin
gs" of algebraic cycles which is a simultaneous generalization of Beilin
son--Bloch height pairing\, and the local intersection pairings \ndevelo
ped by Arakelov\, Faltings\, and Gillet--Soule on Kahler manifolds.
In global setting\,\nour decomposition theorems provide canonical splitt
ings of some canonical filtrations\, including canonical liftings of homo
logical cycles to algebraic cycles.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brendan Creutz (University of Canterbury)
DTSTART:20200929T203000Z
DTEND:20200929T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/9
DESCRIPTION:Title: Quadratic points on del Pezzo surfaces of degree 4\nby Brendan Cr
eutz (University of Canterbury) as part of MIT number theory seminar\n\n\n
Abstract\nI will report on joint work (in progress) with Bianca Viray conc
erning the following question. If $X/k$ is a smooth complete intersection
of $2$ quadrics in $\\mathbb{P}^n$ over a field $k$\, does $X$ have a rati
onal point over some quadratic extension of $k$?\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salim Tayou (Harvard)
DTSTART:20201006T203000Z
DTEND:20201006T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/10
DESCRIPTION:Title: Exceptional jumps of Picard rank of K3 surfaces over number fields\nby Salim Tayou (Harvard) as part of MIT number theory seminar\n\n\nAbs
tract\nGiven a K3 surface X over a number field K\, we prove that the set
of primes of K where the geometric Picard rank jumps is infinite\, assumin
g that X has everywhere potentially good reduction. This result is formula
ted in the general framework of GSpin Shimura varieties and I will explain
other applications to abelian surfaces. I will also discuss applications
to the existence of rational curves on K3 surfaces. The results in this ta
lk are joint work with Ananth Shankar\, Arul Shankar and Yunqing Tang.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UC Santa Barbara)
DTSTART:20201020T203000Z
DTEND:20201020T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/11
DESCRIPTION:Title: Iwasawa theory of elliptic curves at Eisenstein primes and applicati
ons\nby Francesc Castella (UC Santa Barbara) as part of MIT number the
ory seminar\n\n\nAbstract\nIn the study of Iwasawa theory of elliptic curv
es $E/\\mathbb{Q}$\, it is often assumed that $p$ is a non-Eisenstein prim
e\, meaning that $E[p]$ is irreducible as a $G_{\\mathbb{Q}}$-module. Beca
use of this\, most of the recent results on the $p$-converse to the theore
m of Gross–Zagier and Kolyvagin (following Skinner and Wei Zhang) and on
the $p$-part of the Birch–Swinnerton-Dyer formula in analytic rank 1 (f
ollowing Jetchev–Skinner–Wan) were only known for non-Eisenstein prime
s $p$. In this talk\, I’ll explain some of the ingredients in a joint wo
rk with Giada Grossi\, Jaehoon Lee\, and Christopher Skinner in which we s
tudy the (anticyclotomic) Iwasawa theory of elliptic curves over $\\mathbb
{Q}$ at Eisenstein primes. As a consequence of our study\, we obtain an ex
tension of the aforementioned results to the Eisenstein case. In particula
r\, for $p=3$ this leads to an improvement on the best known results towar
ds Goldfeld’s conjecture in the case of elliptic curves over $\\mathbb{Q
}$ with a rational $3$-isogeny.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winnie Li (Pennsylvania State University)
DTSTART:20201027T203000Z
DTEND:20201027T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/12
DESCRIPTION:Title: Pair arithmetical equivalence for quadratic fields\nby Winnie Li
(Pennsylvania State University) as part of MIT number theory seminar\n\n\
nAbstract\nGiven two distinct number fields $K$ and $M$\, and two finite o
rder Hecke characters $\\chi$ of $K$ and $\\eta$ of $M$ respectively\, we
say that the pairs $(\\chi\, K)$ and $(\\eta\, M)$ are arithmetically equi
valent if the associated L-functions coincide: $L(s\, \\chi\, K) = L(s\, \
\eta\, M)$. When the characters are trivial\, this reduces to the question
of fields with the same Dedekind zeta function\, investigated by Gassmann
in 1926\, who found such fields of degree 180\, and by Perlis in 1977 and
others\, who showed that there are no nonisomorphic fields of degree less
than 7.\n\nIn this talk we discuss arithmetically equivalent pairs where
the fields are quadratic. They give rise to dihedral automorphic forms ind
uced from characters of different quadratic fields. We characterize when a
given pair is arithmetically equivalent to another pair\, explicitly cons
truct such pairs for infinitely many quadratic extensions with odd class n
umber\, and classify such characters of order 2.\n\nThis is a joint work w
ith Zeev Rudnick.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ana Caraiani (Imperial College London)
DTSTART:20201103T153000Z
DTEND:20201103T163000Z
DTSTAMP:20260404T094147Z
UID:MITNT/13
DESCRIPTION:Title: Vanishing theorems for Shimura varieties\nby Ana Caraiani (Imper
ial College London) as part of MIT number theory seminar\n\n\nAbstract\nTh
e Langlands program is a vast network of conjectures that connect number t
heory to other areas of mathematics\, such as representation theory and ha
rmonic analysis. The global Langlands correspondence can often be realised
through the cohomology of Shimura varieties\, which are certain moduli sp
aces equipped with many symmetries. In this talk\, I will survey some rece
nt vanishing results for the cohomology of Shimura varieties with mod $p$
coefficients and mention several applications to the Langlands program and
beyond. I will discuss some results that have an $\\ell$-adic flavour\
, where $\\ell$ is a prime different from $p$\, that are primarily joint w
ork with Peter Scholze. I will then mention some results that have a $p$-
adic flavour\, that are primarily joint work with Dan Gulotta and Chris
tian Johansson. I will highlight the different kinds of techniques that ar
e needed in these different settings using the toy model of the modular cu
rve.\n\nThere are two papers that contain work related to this talk: arXiv:1909.01898 and arXiv:1910.0914.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cong Xue (CNRS and IMJ-PRG)
DTSTART:20201110T153000Z
DTEND:20201110T163000Z
DTSTAMP:20260404T094147Z
UID:MITNT/14
DESCRIPTION:Title: Smoothness of the cohomology sheaves of stacks of shtukas\nby Co
ng Xue (CNRS and IMJ-PRG) as part of MIT number theory seminar\n\n\nAbstra
ct\nLet $X$ be a smooth projective geometrically connected curve over a fi
nite field $\\mathbb{F}_q$. Let $G$ be a connected reductive group over th
e function field of $X$. For every finite set $I$ and every representation
of $(\\check{G})^I$\, where $\\check{G}$ is the Langlands dual group of $
G$\, we have a stack of shtukas over $X^I$. For every degree\, we have a c
ompact support $\\ell$-adic cohomology sheaf over $X^I$.\n\nIn this talk\,
I will recall some properties of these sheaves. I will talk about a work
in progress which proves that these sheaves are ind-smooth over $X^I$.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke University)
DTSTART:20201117T213000Z
DTEND:20201117T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/15
DESCRIPTION:Title: Average of $3$-torsion in class groups of $2$-extensions\nby Jiu
ya Wang (Duke University) as part of MIT number theory seminar\n\n\nAbstra
ct\nIn 1971\, Davenport and Heilbronn proved the celebrated theorem determ
ining the average of $3$-torsion in class groups of quadratic extensions.
In this talk\, we will study the average of $3$-torsion in class groups of
$2$-extensions\, which are towers of relative quadratic extensions. As an
example\, we determine the average of $3$-torsion in class groups of $D_4
$ quartic extensions. This is a joint work with Robert J. Lemke Oliver and
Melanie Matchett Wood.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiannis Sakellaridis (Johns Hopkins University)
DTSTART:20210216T213000Z
DTEND:20210216T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/16
DESCRIPTION:Title: Periods\, L-functions\, and duality of Hamiltonian spaces\nby Yi
annis Sakellaridis (Johns Hopkins University) as part of MIT number theory
seminar\n\n\nAbstract\nThe relationship between periods of automorphic fo
rms and L-functions has been studied since the times of Riemann\, but rema
ins mysterious. In this talk\, I will explain how periods and L-functions
arise as quantizations of certain Hamiltonian spaces\, and will propose a
conjectural duality between certain Hamiltonian spaces for a group $G$\, a
nd its Langlands dual group $\\check G$\, in the context of the geometric
Langlands program\, recovering known and conjectural instances of the afor
ementioned relationship. This is joint work with David Ben-Zvi and Akshay
Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lola Thompson (Utrecht University)
DTSTART:20201208T213000Z
DTEND:20201208T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/17
DESCRIPTION:Title: Summing $\\mu(n)$: an even faster elementary algorithm\nby Lola
Thompson (Utrecht University) as part of MIT number theory seminar\n\n\nAb
stract\nWe present a new-and-improved elementary algorithm for computing $
M(x) = \\sum_{n \\leq x} \\mu(n)\,$ where $\\mu(n)$ is the Moebius functio
n. Our algorithm takes time $O\\left(x^{\\frac{3}{5}} \\log \\log x \\righ
t)$ and space $O\\left(x^{\\frac{3}{10}} \\log x \\right)$\, which improv
es on existing combinatorial algorithms. While there is an analytic algori
thm due to Lagarias-Odlyzko with computations based\non integrals of $\\ze
ta(s)$ that only takes time $O(x^{1/2 + \\epsilon})$\, our algorithm has t
he advantage of being easier to implement. The new approach roughly amount
s to analyzing the difference between a model that we obtain via Diophanti
ne approximation and reality\, and showing that it has a simple descriptio
n in terms of congruence classes and segments. This simple description all
ows us to compute the difference quickly by means of a table lookup. This
talk is based on joint work with Harald Andres Helfgott.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke University)
DTSTART:20201215T213000Z
DTEND:20201215T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/18
DESCRIPTION:Title: On superorthogonality\nby Lillian Pierce (Duke University) as pa
rt of MIT number theory seminar\n\n\nAbstract\nThe Burgess bound is a well
-known upper bound for short multiplicative character sums\, which implies
for example a subconvexity bound for Dirichlet L-functions. Since the 195
0's\, people have tried to improve the Burgess method. In order to try to
improve a method\, it makes sense to understand the bigger “proofscape
” in which a method fits. The Burgess method didn’t seem to fit well i
nto a bigger proofscape. In this talk we will show that in fact it can be
regarded as an application of “superorthogonality.” This perspective l
inks topics from harmonic analysis and number theory\, such as Khintchine
’s inequality\, Walsh-Paley series\, square function estimates and decou
pling\, multi-correlation sums of trace functions\, and the Burgess method
. We will survey these connections in an accessible way\, with a focus on
the number theoretic side.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lue Pan (University of Chicago)
DTSTART:20201124T213000Z
DTEND:20201124T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/19
DESCRIPTION:Title: On the locally analytic vectors of the completed cohomology of modul
ar curves\nby Lue Pan (University of Chicago) as part of MIT number th
eory seminar\n\n\nAbstract\nA classical result identifies holomorphic modu
lar forms with\nhighest weight vectors of certain representations of $SL_2
(\\mathbb{R})$. We\nstudy locally analytic vectors of the (p-adically) com
pleted cohomology of\nmodular curves and prove a p-adic analogue of this r
esult. As\napplications\, we are able to prove a classicality result for\n
overconvergent eigenforms and give a new proof of Fontaine-Mazur\nconjectu
re in the irregular case under some mild hypothesis. One technical\ntool i
s relative Sen theory which allows us to relate infinitesimal group\nactio
n with Hodge(-Tate) structure.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Nelson (ETH Zurich)
DTSTART:20210223T153000Z
DTEND:20210223T163000Z
DTSTAMP:20260404T094147Z
UID:MITNT/20
DESCRIPTION:Title: The orbit method\, microlocal analysis and applications to L-functio
ns\nby Paul Nelson (ETH Zurich) as part of MIT number theory seminar\n
\n\nAbstract\nI will describe how the orbit method can be developed in a q
uantitative form\, along the lines of microlocal analysis\, and applied to
local problems in representation theory and global problems concerning au
tomorphic forms. The local applications include asymptotic expansions of
relative characters. The global applications include moment estimates and
subconvex bounds for L-functions. These results are the subject of two p
apers\, the first joint with Akshay Venkatesh:\n\nhttps://arxiv.org/abs/18
05.07750\n\nhttps://arxiv.org/abs/2012.02187\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huajie Li (Aix-Marseille Université)
DTSTART:20210302T153000Z
DTEND:20210302T163000Z
DTSTAMP:20260404T094147Z
UID:MITNT/21
DESCRIPTION:Title: An infinitesimal variant of Guo-Jacquet trace formulae and its compa
rison\nby Huajie Li (Aix-Marseille Université) as part of MIT number
theory seminar\n\n\nAbstract\nThe Guo-Jacquet conjecture is a promising ge
neralization to higher dimensions of Waldspurger’s well-known theorem re
lating toric periods to central values of automorphic L-functions for $GL(
2)$. Feigon-Martin-Whitehouse have proved some cases of this conjecture us
ing simple relative trace formulae\, Guo’s work on the fundamental lemma
and C. Zhang’s work on the transfer. However\, if we want to obtain mor
e general results\, we have to establish and compare more general relative
trace formulae\, where some analytic difficulties such as the divergence
issue should be addressed. \n\nIn this talk\, we plan to study analogues o
f these problems at the infinitesimal level. After briefly introducing the
background\, we shall present an infinitesimal variant of Guo-Jacquet tra
ce formulae. To compare regular semi-simple terms in these formulae\, we s
hall discuss the weighted fundamental lemma and certain identities between
Fourier transforms of local weighted orbital integrals. During the proof\
, we also need some results in local harmonic analysis such as local trace
formulae for some $p$-adic infinitesimal symmetric spaces. This talk is b
ased on my thesis supervised by P.-H. Chaudouard.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Gleason (UC Berkeley)
DTSTART:20210309T213000Z
DTEND:20210309T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/22
DESCRIPTION:Title: On the geometric connected components of moduli of p-adic shtukas.\nby Ian Gleason (UC Berkeley) as part of MIT number theory seminar\n\n\
nAbstract\nThrough the recent theory of diamonds\, P. Scholze constructs l
ocal Shimura varieties and moduli of p-adic shtukas attached to any reduct
ive group. These are diamonds that generalize the generic fiber of a Rapop
ort–Zink space. These interesting spaces realize in their cohomology ins
tances of the local Langlands correspondence. In this talk\, we describe t
he set of connected components of moduli spaces of p-adic shtukas (with on
e paw). The new ingredient of this work is the use of specialization maps
in the context of diamonds.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Zerbes (University College London)
DTSTART:20210316T143000Z
DTEND:20210316T153000Z
DTSTAMP:20260404T094147Z
UID:MITNT/23
DESCRIPTION:Title: Euler systems and explicit reciprocity laws for GSp(4)\nby Sarah
Zerbes (University College London) as part of MIT number theory seminar\n
\n\nAbstract\nEuler systems are a very powerful tool for attacking the Blo
ch—Kato conjecture\, which is one of the central open problems in number
theory. In this talk\, I will sketch the construction of an Euler system
for the spin Galois representation of a genus 2 Siegel modular form. I wil
l then explain how to prove an explicit reciprocity law\, relating the ima
ge of the Euler system under the Bloch—Kato logarithm map to values of t
he complex L-function of the Siegel modular form. The applications of this
result to the Bloch—Kato conjecture and the Iwasawa Main Conjecture wil
l be discussed by David Loeffler in the following week.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Loeffler (University of Warwick)
DTSTART:20210323T143000Z
DTEND:20210323T153000Z
DTSTAMP:20260404T094147Z
UID:MITNT/24
DESCRIPTION:Title: The Bloch--Kato conjecture for critical values of GSp(4) L-functions
\nby David Loeffler (University of Warwick) as part of MIT number theo
ry seminar\n\n\nAbstract\nIn Sarah's talk last week\, she explained the co
nstruction of a family of Galois cohomology \nclasses (an Euler system) at
tached to Siegel modular forms\, and related the localisations of these cl
asses at p to non-critical values of p-adic L-functions. In this talk\, I
will explain how to 'analytically continue' this relation to obtain an exp
licit reciprocity law relating Galois cohomology classes to critical value
s of L-functions\; and I will discuss applications of this result to the B
loch--Kato conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Teppei Takamatsu (University of Tokyo)
DTSTART:20210330T203000Z
DTEND:20210330T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/25
DESCRIPTION:Title: Minimal model program for semi-stable threefolds in mixed characteri
stic\nby Teppei Takamatsu (University of Tokyo) as part of MIT number
theory seminar\n\n\nAbstract\nThe minimal model program\, which is a theor
y to construct a birational model of a variety which is as simple as possi
ble\, is a very strong method in algebraic geometry.\nThe minimal model pr
ogram is also studied for more general schemes not necessarily defined o
ver a field\, and play an important role in studies of reductions of varie
ties.\nKawamata showed that the minimal model program holds for strictly s
emi-stable schemes over an excellent Dedekind scheme of relative dimensi
on two whose each residue characteristic is neither 2 nor 3.\nIn this talk
\, I will introduce a generalization of the result of Kawamata without an
y assumption on the residue characteristic.\nThis talk is based on a joint
work with Shou Yoshikawa.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART:20210406T143000Z
DTEND:20210406T153000Z
DTSTAMP:20260404T094147Z
UID:MITNT/26
DESCRIPTION:Title: Almost primes in almost all very short intervals\nby Kaisa Matom
äki (University of Turku) as part of MIT number theory seminar\n\n\nAbstr
act\nBy probabilistic models one expects that\, as soon as $h \\to \\infty
$ with $X \\to \\infty$\, short intervals of the type $(x- h \\log X\, x]$
contain primes for almost all $x \\in (X/2\, X]$. However\, this is far f
rom being established. In the talk I discuss related questions and in part
icular describe how to prove the above claim when one is satisfied with fi
nding $P_2$-numbers (numbers that have at most two prime factors) instead
of primes.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pol van Hoften (King's College London)
DTSTART:20210413T143000Z
DTEND:20210413T153000Z
DTSTAMP:20260404T094147Z
UID:MITNT/27
DESCRIPTION:Title: Mod $p$ points on Shimura varieties of parahoric level\nby Pol v
an Hoften (King's College London) as part of MIT number theory seminar\n\n
\nAbstract\nThe conjecture of Langlands-Rapoport gives a conjectural descr
iption of the mod $p$ points of Shimura varieties\, with applications towa
rds computing the (semi-simple) zeta function of these Shimura varieties.
The conjecture was proven by Kisin for abelian type Shimura varieties at p
rimes of (hyperspecial) good reduction\, after having constructed smooth i
ntegral models. For primes of (parahoric) bad reduction\, Kisin and Pappas
have constructed a good integral model and the conjecture was generalised
to this setting by Rapoport. In this talk I will discuss recent results t
owards the conjecture for these integral models\, under minor hypothesis\,
building on earlier work of Zhou. Along the way we will see irreducibilit
y results for various stratifications on special fibers of Shimura varieti
es\, including irreducibility of central leaves and Ekedahl-Oort strata.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ila Varma (University of Toronto)
DTSTART:20210427T203000Z
DTEND:20210427T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/28
DESCRIPTION:Title: Malle's conjecture for octic $D_4$-fields.\nby Ila Varma (Univer
sity of Toronto) as part of MIT number theory seminar\n\n\nAbstract\nWe co
nsider the family of normal octic fields with Galois group $D_4$\, ordered
by their discriminant. In forthcoming joint work with Arul Shankar\, we v
erify the strong form of Malle's conjecture for this family of number fiel
ds\, obtaining the order of growth as well as the constant of proportional
ity. In this talk\, we will discuss and review the combination of techniqu
es from analytic number theory and geometry-of-numbers methods used to pro
ve this and related results.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Bayer-Fluckiger (EPFL)
DTSTART:20210504T143000Z
DTEND:20210504T153000Z
DTSTAMP:20260404T094147Z
UID:MITNT/29
DESCRIPTION:Title: Isometries of lattices and Hasse principle\nby Eva Bayer-Fluckig
er (EPFL) as part of MIT number theory seminar\n\n\nAbstract\nWe give nece
ssary and sufficient conditions for an integral polynomial to be the chara
cteristic polynomial of an isometry of some even\, unimodular lattice of g
iven signature.\n\nRelated papers: arXiv:2001.07094\, arXi
v:2107.07583.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugen Hellmann (Mathematisches Institut Münster)
DTSTART:20210511T143000Z
DTEND:20210511T153000Z
DTSTAMP:20260404T094147Z
UID:MITNT/30
DESCRIPTION:Title: On classicality of overconvergent $p$-adic automorphic forms\nby
Eugen Hellmann (Mathematisches Institut Münster) as part of MIT number t
heory seminar\n\n\nAbstract\nI will report on some positive and a negative
result concerning the question whether a given overconvergent $p$-adic ei
genform of finite slope is classical or not. \nThe positive result is the
generalization of a classicality statement (obtained in earlier joint work
with Breuil and Schraen) to the case of semi-stable Galois representation
s. This classicality result is rather a statement about the Galois represe
ntation attached to a $p$-adic automorphic form than a statement about the
$p$-adic automorphic form itself. The negative result concerns the classi
cality problem for the $p$-adic automorphic form itself. If time permits w
e will discuss some conjectural picture explaining this negative result.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART:20210921T203000Z
DTEND:20210921T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/31
DESCRIPTION:Title: Abelian Varieties with given $p$-torsion representations\nby Shi
va Chidambaram (MIT) as part of MIT number theory seminar\n\nLecture held
in Room 2-143 in the Simons building (building 2).\n\nAbstract\nThe Siegel
modular variety $\\mathcal{A}_2(3)$\, which parametrizes abelian surfaces
with full level $3$ structure\, was shown to be rational over $\\Q$ by Br
uin and Nasserden. What can we say about its twist $\\mathcal{A}_2(\\rho)$
\, parametrizing abelian surfaces $A$ with $\\rho_{A\,3} \\simeq \\rho$\,
for a given mod $3$ Galois representation $\\rho : G_{\\Q} \\rightarrow \\
GSp(4\, \\F_3)$? While it is not rational in general\, it is unirational o
ver $\\Q$ by a map of degree at most $6$\, if $\\rho$ satisfies the necess
ary condition of having cyclotomic similitude. In joint work with Frank Ca
legari and David Roberts\, we obtain an explicit description of the univer
sal object over a degree $6$ cover of $\\mathcal{A}_2(\\rho)$\, using inva
riant theoretic ideas. One application of this result is towards an explic
it transfer of modularity\, yielding infinitely many examples of modular a
belian surfaces with no extra endomorphisms. Similar ideas work in a few o
ther cases\, showing in particular that whenever $(g\,p) = (1\,2)$\, $(1\,
3)$\, $(1\,5)$\, $(2\,2)$\, $(2\,3)$ and $(3\,2)$\, the cyclotomic similit
ude condition is also sufficient for a mod $p$ Galois representation to ar
ise from the $p$-torsion of a $g$-dimensional abelian variety. When $(g\,p
)$ is not one of these six tuples\, we will discuss a local obstruction fo
r representations to arise as torsion.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART:20210928T203000Z
DTEND:20210928T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/32
DESCRIPTION:Title: Quadratic points on intersections of quadrics\nby Bianca Viray (
University of Washington) as part of MIT number theory seminar\n\nLecture
held in Room 2-143 in the Simons building (building 2).\n\nAbstract\nA pro
jective degree $d$ variety always has a point defined over a degree $d$ fi
eld extension. For many degree $d$ varieties\, this is the best possible
statement\, that is\, there exist classes of degree $d$ varieties that nev
er have points over extensions of degree less than $d$ (nor even over exte
nsions whose degree is nonzero modulo $d$). However\, there are some clas
ses of degree $d$ varieties that obtain points over extensions of smaller
degree\, for example\, degree $9$ surfaces in $\\mathbb{P}^9$\, and $6$-di
mensional intersections of quadrics over local fields. In this talk\, we
explore this question for intersections of quadrics. In particular\, we p
rove that a smooth complete intersection of two quadrics of dimension at l
east $2$ over a number field has index dividing $2$\, i.e.\, that it posse
sses a rational $0$-cycle of degree $2$. This is joint work with Brendan
Creutz.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Kieffer (Harvard)
DTSTART:20211005T203000Z
DTEND:20211005T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/33
DESCRIPTION:Title: Higher-dimensional modular equations and point counting on abelian s
urfaces\nby Jean Kieffer (Harvard) as part of MIT number theory semina
r\n\nLecture held in Room 2-143 in the Simons building (building 2).\n\nAb
stract\nGiven a prime number $\\ell$\, the elliptic modular polynomial of
level\n$\\ell$ is an explicit equation for the locus of elliptic curves\nr
elated by an $\\ell$-isogeny. These polynomials have a large number of\nal
gorithmic applications: in particular\, they are an essential\ningredient
in the celebrated SEA algorithm for counting points on\nelliptic curves ov
er finite fields of large characteristic.\n\nMore generally\, modular equa
tions describe the locus of isogenous\nabelian varieties in certain moduli
spaces called PEL Shimura\nvarieties. We will present upper bounds on the
size of modular\nequations in terms of their level\, and outline a genera
l strategy to\ncompute an isogeny $A\\to A'$ given a point $(A\,A')$ where
the modular\nequations are satisfied. This generalizes well-known propert
ies of\nelliptic modular polynomials to higher dimensions.\n\nThe isogeny
algorithm is made fully explicit for Jacobians of genus 2\ncurves. In comb
ination with efficient evaluation methods for modular\nequations in genus
2 via complex approximations\, this gives rise to\npoint counting algorith
ms for (Jacobians of) genus 2 curves whose\nasymptotic costs\, under stand
ard heuristics\, improve on previous\nresults.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Kisin (Harvard)
DTSTART:20211014T190000Z
DTEND:20211014T200000Z
DTSTAMP:20260404T094147Z
UID:MITNT/34
DESCRIPTION:Title: Essential dimension via prismatic cohomology\nby Mark Kisin (Har
vard) as part of MIT number theory seminar\n\nLecture held in Room 2-449 i
n the Simons building.\n\nAbstract\nLet $f:Y \\rightarrow X$ be a finite c
overing map of complex algebraic varieties. The essential dimension of $f$
is the smallest integer $e$ such that\, birationally\, $f$ arises as the
pullback \nof a covering $Y' \\rightarrow X'$ of dimension $e\,$ via a map
$X \\rightarrow X'.$ This invariant goes back to classical questions abou
t reducing the number of parameters in a solution to a general $n^{\\rm th
}$ degree polynomial\, and appeared in work of Kronecker and Klein on solu
tions of the quintic. \n\nI will report on joint work with Benson Farb and
Jesse Wolfson\, where we introduce a new technique\, using prismatic coho
mology\, to obtain lower bounds on the essential dimension of certain cove
rings. For example\, we show that for an abelian variety $A$ of dimension
$g$ the multiplication by $p$ map $A \\rightarrow A$ has essential dimensi
on $g$ for almost all primes $p.$\n\nNote the unusual time and place: Thur
sday at 3pm in 2-449.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Myrto Mavraki (Harvard)
DTSTART:20211026T203000Z
DTEND:20211026T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/36
DESCRIPTION:Title: On the dynamical Bogomolov conjecture\nby Myrto Mavraki (Harvard
) as part of MIT number theory seminar\n\nLecture held in Room 2-143 in th
e Simons building.\n\nAbstract\nMotivated by the Manin-Mumford conjecture\
, established by Raynaud\, and following the analogy of torsion with prepe
riodic points\, Zhang posed a dynamical Manin-Mumford conjecture. Using a
canonical height introduced by Call and Silverman he further formulated a
dynamical Bogomolov conjecture. A special case of these conjectures has re
cently been established by Nguyen\, Ghioca and Ye. In particular\, they sh
ow that two rational maps have at most finitely many common preperiodic po
ints\, unless they are 'related'. In this talk we discuss relative and uni
form versions of such results. This is joint work with Harry Schmidt.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Landesman (Harvard)
DTSTART:20211102T203000Z
DTEND:20211102T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/37
DESCRIPTION:Title: The geometric distribution of Selmer groups of Elliptic curves over
function fields\nby Aaron Landesman (Harvard) as part of MIT number th
eory seminar\n\nLecture held in Room 2-143 in the Simons building.\n\nAbst
ract\nBhargava\, Kane\, Lenstra\, Poonen\, and Rains proposed heuristics f
or the distribution of arithmetic data relating to elliptic curves\, such
as their ranks\, Selmer groups\, and Tate-Shafarevich groups.\nAs a specia
l case of their heuristics\, they obtain the minimalist conjecture\, which
predicts that $50\\%$ of elliptic curves have rank $0$ and $50\\%$ of ell
iptic curves have rank $1$. \nAfter surveying these conjectures\, we will
explain joint work with Tony Feng and Eric Rains\, \nverifying a variant o
f these conjectures over function fields of the form $\\mathbb F_q(t)$\, a
fter taking a certain large $q$ limit.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Harvard)
DTSTART:20211109T213000Z
DTEND:20211109T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/38
DESCRIPTION:Title: Finitely Presented Groups in Arithmetic Geometry\nby Mark Shuste
rman (Harvard) as part of MIT number theory seminar\n\nLecture held in Roo
m 2-143 in the Simons building.\n\nAbstract\nWe discuss the problem of det
ermining the number of generators and relations of several profinite group
s of arithmetic and geometric origin. \nThese include etale fundamental gr
oups of smooth projective varieties\, absolute Galois groups of local fiel
ds\, and Galois groups of maximal unramified extensions of number fields.
The results are based on a cohomological presentability criterion of Lubot
zky\, and draw inspiration from well-known facts about three-dimensional m
anifolds\, as in arithmetic topology. \n\nThe talk is based in part on c
ollaborations with Esnault\, Jarden\, and Srinivas.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpöge (Harvard)
DTSTART:20211116T213000Z
DTEND:20211116T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/39
DESCRIPTION:Title: A "height-free" effective isogeny estimate for abelian varieties of
$\\GL_2$-type.\nby Levent Alpöge (Harvard) as part of MIT number theo
ry seminar\n\nLecture held in Room 2-143 in the Simons building.\n\nAbstra
ct\nLet $g\\in \\mathbb{Z}^+$\, $K$ a number field\, $S$ a finite set of p
laces of $K$\, and $A\,B/K$ $g$-dimensional abelian varieties with good re
duction outside $S$ which are $K$-isogenous and of $\\GL_2$-type over $\\o
verline{\\mathbb{Q}}$. We show that there is a $K$-isogeny $A\\rightarrow
B$ of degree effectively bounded in terms of $g$\, $K$\, and $S$ only.\n\n
We deduce among other things an effective upper bound on the number of $S$
-integral $K$-points on a Hilbert modular variety.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Graham (Université Paris-Saclay)
DTSTART:20220215T213000Z
DTEND:20220215T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/40
DESCRIPTION:Title: Functoriality of higher Coleman theory and p-adic L-functions in the
unitary setting\nby Andrew Graham (Université Paris-Saclay) as part
of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons B
uilding (building 2).\n\nAbstract\nI will describe the construction of a p
-adic analytic function interpolating unitary Friedberg--Jacquet periods\,
which are conjecturally related to central critical values of L-functions
for cuspidal automorphic representations of unitary groups. The construct
ion involves establishing functoriality of Boxer and Pilloni's higher Cole
man theory\, and p-adically interpolating branching laws for a certain pai
r of unitary groups. The motivation for such a p-adic analytic function ar
ises from the Bloch--Kato conjecture for twists of the associated Galois r
epresentation by anticyclotomic characters.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Krieger (University of Cambridge)
DTSTART:20220222T213000Z
DTEND:20220222T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/41
DESCRIPTION:Title: Transcendental values of power series and dynamical degrees\nby
Holly Krieger (University of Cambridge) as part of MIT number theory semin
ar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nA
bstract\nI will explain the construction (joint with Bell\, Diller\, and J
onsson) of a birational map of projective 3-space with transcendental dyna
mical degree. This number is a measure of algebraic complexity of the ite
rates of a rational map\, and was previously conjectured to be algebraic f
or all birational maps. Our proof includes a more general statement on tr
anscendental values of certain power series\, using techniques similar to
those of Adamczewski-Bugeaud.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keerthi Madapusi (Boston College)
DTSTART:20220308T213000Z
DTEND:20220308T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/43
DESCRIPTION:Title: Derived special cycles on Shimura varieties\nby Keerthi Madapusi
(Boston College) as part of MIT number theory seminar\n\nLecture held in
Room 2-449 in the Simons Building (building 2).\n\nAbstract\nWe employ met
hods from derived algebraic geometry to give a uniform moduli-theoretic co
nstruction of special cycle classes on many Shimura varieties of Hodge typ
e. Our results apply in particular to classes on GSpin Shimura varieties a
ssociated with arbitrary positive semi-definite symmetric matrices\, as we
ll as to certain unitary and quaternionic Shimura varieties. We show that
these classes agree with the ones constructed in work with B. Howard using
$K$-theoretic methods.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Silverman (Brown University)
DTSTART:20220412T203000Z
DTEND:20220412T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/47
DESCRIPTION:Title: Orbits on tri-involutive K3 surfaces\nby Joseph Silverman (Brown
University) as part of MIT number theory seminar\n\nLecture held in Room
2-449 in the Simons Building (building 2).\n\nAbstract\nLet $W$ be a surfa
ce in $\\mathbb{P}^1 \\times \\mathbb{P}^1 \\times \\mathbb{P}^1$ given by
the vanishing of a $(2\,2\,2)$ form. The three projections $W \\to \\math
bb{P}^1 \\times \\mathbb{P}^1$ are double covers that induce three non-co
mmuting involutions on $W$. Let $G$ be the group of automorphisms of $W$
generated by these involutions. We investigate the $G$-orbit structure of
the points of $W$. In particular\, we study $G$-orbital components over fi
nite fields and finite $G$-orbits in characteristic 0. This is joint work
with Elena Fuchs\, Matthew Litman\, and Austin Tran.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Harvard)
DTSTART:20220419T203000Z
DTEND:20220419T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/48
DESCRIPTION:Title: Galois action on the pro-algebraic fundamental group\nby Alexand
er Petrov (Harvard) as part of MIT number theory seminar\n\nLecture held i
n Room 2-449 in the Simons Building (building 2).\n\nAbstract\nGiven a smo
oth variety X over a number field\, the action of the Galois group on the
geometric etale fundamental group of X makes the ring of functions on the
pro-algebraic completion of this fundamental group into a (usually infinit
e-dimensional) Galois representation. This Galois representation turns out
to satisfy the following two properties:\n\n1)Every finite-dimensional su
brepresentation of it satisfies the assumptions of the Fontaine-Mazur conj
ecture: it is de Rham an almost everywhere unramifed.\n\n2)If X is the pro
jective line with three punctures\, the semi-simplification of every Galoi
s representation of geometric origin is a subquotient of the ring of regul
ar functions on the pro-algebraic completion of the etale fundamental grou
p of X.\n\nI will also discuss a conjectural characterization of local sys
tems of geometric origin on complex algebraic varieties\, arising from pro
perty 1) above.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melanie Matchett Wood (Harvard)
DTSTART:20220426T203000Z
DTEND:20220426T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/49
DESCRIPTION:Title: Distributions of unramified extensions of global fields\nby Mela
nie Matchett Wood (Harvard) as part of MIT number theory seminar\n\nLectur
e held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nEve
ry number field K has a maximal unramified extension K^un\, with\nGalois g
roup Gal(K^un/K) (whose abelianization is the class group of\nK). As K v
aries\, we ask about the distribution of the groups\nGal(K^un/K). We pro
ve some results about the structure of Gal(K^un/K) \nthat motivate us to g
ive a conjecture about this distribution\, which we\nalso conjecture in th
e function field analog. We give theorems in\nthe function field case (a
s the size of the finite field goes to\ninfinity) that support these new c
onjectures. In particular\, our\ndistributions abelianize to the Cohen-L
enstra-Martinet distributions\nfor class groups\, and so our function fiel
d theorems prove\n(suitably modified) versions of the Cohen-Lenstra-Martin
et heuristics\nover function fields as the size of the finite field goes t
o\ninfinity. This talk is on joint work with Yuan Liu and David Zureick-
Brown.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Pollack (Boston University)
DTSTART:20220503T203000Z
DTEND:20220503T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/50
DESCRIPTION:Title: Predicting slopes of modular forms and reductions of crystalline rep
resentations\nby Robert Pollack (Boston University) as part of MIT num
ber theory seminar\n\nLecture held in Room 2-449 in the Simons Building (b
uilding 2).\n\nAbstract\nThe ghost conjecture predicts slopes of modular f
orms whose residual representation is locally reducible. In this talk\, w
e'll examine locally irreducible representations and discuss recent progre
ss on formulating a conjecture in this case. It's a lot trickier and the
story remains incomplete\, but we will discuss how an irregular ghost conj
ecture is intimately related to reductions of crystalline representations.
\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noam David Elkies (Harvard)
DTSTART:20220510T203000Z
DTEND:20220510T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/51
DESCRIPTION:Title: Abelian surfaces with a (1\,2) polarization and full level-2 structu
re\nby Noam David Elkies (Harvard) as part of MIT number theory semina
r\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAb
stract\nAbstract: The moduli threefold of principally polarized abelian su
rfaces\nwith full level-2 structure is well understood thanks to its close
\nconnection with the moduli space $M_{0\,6}$ of six points on ${\\bf P}^1
$.\nThe moduli threefolds of (1\,d)-polarized surfaces with d>1 are more e
lusive.\nWe report on our recent work on the d=2 case with full level-2 st
ructure.\nHere the moduli threefold is still rational\, and comes with\nan
action of a group G isomorphic with ${\\rm Aut}(S_4^2)$ instead of $S_6$.
\nWe use elliptic fibrations of the Kummer surface to give\nseveral models
of this moduli threefold together with the G-action.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kimball Martin (University of Oklahoma)
DTSTART:20220315T203000Z
DTEND:20220315T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/52
DESCRIPTION:Title: Quaternionic and algebraic modular forms: structure and applications
\nby Kimball Martin (University of Oklahoma) as part of MIT number the
ory seminar\n\nLecture held in Room 2-449 in the Simons Building (building
2).\n\nAbstract\nModular forms on definite quaternion algebras are amenab
le to exact calculation by algebraic methods\, and are related to classica
l modular forms via the Jacquet-Langlands correspondence. I will describe
some structural results on quaternionic modular forms and applications to
computing modular forms\, Eisenstein congruences and central $L$-values.
Along the way\, I will discuss issues and progress toward analogues for a
lgebraic modular forms on higher rank groups.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiran Kedlaya (University of California San Diego)
DTSTART:20220915T190000Z
DTEND:20220915T200000Z
DTSTAMP:20260404T094147Z
UID:MITNT/53
DESCRIPTION:Title: The relative class number one problem for function fields\nby Ki
ran Kedlaya (University of California San Diego) as part of MIT number the
ory seminar\n\nLecture held in Room 4-145.\n\nAbstract\nBuilding on my lec
ture from ANTS-XV\, we classify extensions of function fields (of curves o
ver finite fields) with relative class number 1. Many of the ingredients c
ome from the study of the maximum number of points on a curve over a finit
e field\, such as the function field analogue of Weil's explicit formulas
(a/k/a the "linear programming method"). Additional tools include the clas
sification of abelian varieties of order 1 and the geometry of moduli spac
es of curves of genus up to 7.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Disegni (Ben-Gurion University of the Negev)
DTSTART:20220920T214500Z
DTEND:20220920T224500Z
DTSTAMP:20260404T094147Z
UID:MITNT/54
DESCRIPTION:Title: Algebraic cycles and p-adic L-functions for conjugate-symplectic mot
ives\nby Daniel Disegni (Ben-Gurion University of the Negev) as part o
f MIT number theory seminar\n\nLecture held in Room 2-143 in the Simons Bu
ilding (building 2).\n\nAbstract\nI will introduce ‘canonical’ algebra
ic cycles for motives $M$ enjoying a certain symmetry - for instance\,
some symmetric powers of elliptic curves. The construction is based on wor
ks of Kudla and Liu on some (conjecturally modular) theta series valued
in Chow groups of Shimura varieties. The cycles have Heegner-point-like fe
atures that allow\, under some assumptions\, to support an analogue of the
BSD conjecture for M at an ordinary prime $p$. Namely: if the $p$-adic $L
$-function of $M$ vanishes at the center to order exactly 1\, then the ${\
\bf Q}_p$-Selmer group of $M$ has rank 1\, and it is generated by classes
of algebraic cycles. Partly joint work with Yifeng Liu.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Calegari (University of Chicago)
DTSTART:20220927T203000Z
DTEND:20220927T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/55
DESCRIPTION:Title: The arithmetic of power series\nby Frank Calegari (University of
Chicago) as part of MIT number theory seminar\n\nLecture held in Room 2-1
43 in the Simons Building (building 2).\n\nAbstract\nAbstract: A function
of a complex variable $P(z)$ which is holomorphic around $z=0$ has a power
series expansion $P(z)=\\sum a_n z^n$. Suppose that the $a_n$ are all int
egers: what restrictions does that place on the function $P(z)$? We explor
e the relationship between this problem to questions in complex analysis\,
number theory\, and to Klein’s famous observation that not all finite i
ndex subgroups of $\\mathrm{SL}_2(\\mathbf{Z})$ are determined by congruen
ce conditions. This talk is based on joint work with Vesselin Dimitrov and
Yunqing Tang.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (MIT)
DTSTART:20221004T203000Z
DTEND:20221004T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/56
DESCRIPTION:Title: Normalization in the integral models of Shimura varieties of abelian
type\nby Yujie Xu (MIT) as part of MIT number theory seminar\n\nLectu
re held in Room 2-143 in the Simons Building (building 2).\n\nAbstract\nSh
imura varieties are moduli spaces of abelian varieties with extra structur
es. Many interesting questions about abelian varieties have been answered
by studying the geometry of Shimura varieties. \n\nIn order to study the m
od $p$ points of Shimura varieties\, over the decades\, various mathematic
ians (e.g. Rapoport\, Kottwitz\, etc.) have constructed nice integral mode
ls of Shimura varieties. \nIn this talk\, I will discuss some motivic aspe
cts of integral models of Hodge type (or more generally abelian type) cons
tructed by Kisin and Kisin-Pappas. I will talk about my recent work on rem
oving the normalization step in the construction of such integral models\,
which gives closed embeddings of Hodge type integral models into Siegel i
ntegral models. I will also mention an application to toroidal compactific
ations of such integral models. Such results (and their proof techniques)
have found interesting applications to the Kudla program (and various othe
r programs!).\n\nIf time permits\, I will also mention a new result on con
nected components of affine Deligne–Lusztig varieties\, which gives us a
new CM lifting result for integral models of Shimura varieties at parahor
ic levels and serves as an ingredient for my main theorem at parahoric lev
els.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashvin Swaminathan (Harvard)
DTSTART:20221011T203000Z
DTEND:20221011T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/57
DESCRIPTION:Title: Geometry-of-numbers in the cusp\, and class groups of orders in numb
er fields\nby Ashvin Swaminathan (Harvard) as part of MIT number theor
y seminar\n\nLecture held in Room 2-143 in the Simons Building (building 2
).\n\nAbstract\nIn this talk\, we discuss the distributions of class group
s of orders in number fields. We explain how studying such distributions i
s related to counting integral orbits having bounded invariants that lie i
nside the cusps of fundamental domains for coregular representations. We i
ntroduce two new methods to solve this counting problem\, and as an exampl
e\, we demonstrate how one of these methods can be used to determine the a
verage size of the 2-torsion in the class groups of totally real or comple
x cubic orders\, when such orders are enumerated by discriminant. Much of
this work is joint with Arul Shankar\, Artane Siad\, and Ila Varma.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Rüd (MIT)
DTSTART:20221018T203000Z
DTEND:20221018T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/58
DESCRIPTION:Title: Tamagawa numbers for maximal symplectic tori and mass formulae for a
belian varieties\nby Thomas Rüd (MIT) as part of MIT number theory se
minar\n\nLecture held in Room 2-143 in the Simons Building (building 2).\n
\nAbstract\nTamagawa numbers are defined as a specific volumes of algebrai
c group\, encapsulating the size of their adelic points modulo rational po
ints. Unsurprisingly\, such numbers are intrinsically linked to mass formu
lae of various kinds. More recently\, Tamagawa numbers of centralizers of
elements within some algebraic groups also appear in the context of the st
able trace formula.\n\nGekeler's result on a mass formula for elliptic cur
ves defined over $\\mathbb{F}_p$ was extended by Achter-Gordon and then Ac
hter-Altug-Garcia-Gordon to mass formulae for certain principally polarize
d abelian varieties over finite fields\, using orbital integrals appearing
in Langlands-Kottwitz formula. \nGekeler's work uses the analytic class n
umber formula\, which can be restated as $\\tau_\\mathbb{Q}(\\mathrm{R}_{K
/\\mathbb{Q}}\\mathbb{G}_m)=1$ ($\\mathrm{R}$ denotes the Weil restriction
of scalars)\, but in the case of higher-dimensional abelian varieties the
tori involved are more complicated and their Tamagawa numbers were not kn
own.\n\n\nThe work presented aims at showing techniques to compute such nu
mbers as well as many general results on Tamagawa numbers of a vast class
of tori\, which includes maximal tori of $\\mathrm{GSp}_{2n}$ over any glo
bal field. In particular we will give extensive results for tori splitting
over CM-fields and the possible range of such Tamagawa numbers.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Congling Qiu (Yale)
DTSTART:20221025T203000Z
DTEND:20221025T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/59
DESCRIPTION:Title: Arithmetic mixed Siegel-Weil formulas and modular form of arithmetic
divisors\nby Congling Qiu (Yale) as part of MIT number theory seminar
\n\nLecture held in Room 2-143 in the Simons Building (building 2).\n\nAbs
tract\nThe classical Siegel–Weil formula relates theta series to Eise
nstein series and its arithmetic version is central in Kudla's program. I
will discuss arithmetic mixed Siegel-Weil formulas. I will focus on the on
e in the work of Gross and Zagier\, and the one in my recent work. As an a
pplication\, I obtained modular generating series of arithmetic extension
s of Kudla's special divisors for unitary Shimura varieties over CM fields
with arbitrary split level. This provides a partial solution to a problem
of Kudla.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano López (Harvard University)
DTSTART:20221101T203000Z
DTEND:20221101T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/60
DESCRIPTION:Title: Counting fields generated by points on plane curves\nby Allechar
Serrano López (Harvard University) as part of MIT number theory seminar\
n\nLecture held in Room 2-143 in the Simons Building (building 2).\n\nAbst
ract\nFor a smooth projective curve $C/\\mathbb{Q}$\, how many field exten
sions of $\\mathbb{Q}$ -- of given degree and bounded discriminant --- ari
se from adjoining a point of $C(\\overline{\\mathbb{Q}})$? Can we further
count the number of such extensions with a specified Galois group? Asympto
tic lower bounds for these quantities have been found for elliptic curves
by Lemke Oliver and Thorne\, for hyperelliptic curves by Keyes\, and for s
uperelliptic curves by Beneish and Keyes. We discuss similar asymptotic lo
wer bounds that hold for all smooth plane curves $C$. This is joint work w
ith Michael\, Allen\, Renee Bell\, Robert Lemke Oliver\, and Tian An Wong.
\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Cowan (Harvard)
DTSTART:20221108T213000Z
DTEND:20221108T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/61
DESCRIPTION:Title: A twisted additive divisor problem\nby Alex Cowan (Harvard) as p
art of MIT number theory seminar\n\nLecture held in Room 2-143 in the Simo
ns Building (building 2).\n\nAbstract\nWe give asymptotics for a shifted c
onvolution of sum-of-divisors functions twisted by Dirichlet characters an
d with nonzero powers. We'll use the technique of "automorphic regularizat
ion" to find a spectral decomposition of a combination of Eisenstein serie
s which is not obviously square-integrable.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barinder Banwait (Boston University)
DTSTART:20221115T213000Z
DTEND:20221115T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/62
DESCRIPTION:Title: Towards strong uniformity for isogenies of prime degree\nby Bari
nder Banwait (Boston University) as part of MIT number theory seminar\n\nL
ecture held in Room 2-143 in the Simons Building (building 2).\n\nAbstract
\nLet $E$ be an elliptic curve over a number field $k$ of degree $d$ that
admits a $k$-rational isogeny of prime degree $p$. We study the question o
f finding uniform bounds on $p$ that depend only $d$\, and\, under a certa
in condition on the signature of the isogeny\, explicitly construct non-ze
ro integers that $p$ must divide. As a corollary\, we find a bound on prim
e order torsion points defined over unramified extensions of the base fiel
d. This is work in progress joint with Maarten Derickx.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (Stanford)
DTSTART:20221122T213000Z
DTEND:20221122T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/63
DESCRIPTION:Title: Simple abelian varieties over finite fields with extreme point count
s\nby Alexander Smith (Stanford) as part of MIT number theory seminar\
n\nLecture held in Room 2-143 in the Simons Building (building 2).\n\nAbst
ract\nGiven a compactly supported probability measure on the reals\, we wi
ll give a necessary and sufficient condition for there to be a sequence of
totally real algebraic integers whose distribution of conjugates approach
es the measure. We use this result to prove that there are infinitely many
totally positive algebraic integers X satisfying tr(X)/deg(X) < 1.899\; p
reviously\, there were only known to be infinitely many such integers sati
sfying tr(X)/deg(X) < 2. We also will explain how our method can be used i
n the search for simple abelian varieties with extreme point counts.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Álvaro Lozano-Robledo (University of Connecticut)
DTSTART:20221129T213000Z
DTEND:20221129T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/64
DESCRIPTION:Title: The adelic image of a Galois representation attached to a CM ellipti
c curve\nby Álvaro Lozano-Robledo (University of Connecticut) as part
of MIT number theory seminar\n\nLecture held in Room 2-143 in the Simons
Building (building 2).\n\nAbstract\nIn this talk we will discuss recent wo
rk on the classification of $\\ell$-adic images of Galois representations
attached to elliptic curves with complex multiplication\, and applications
. In particular\, we will show how to construct the adelic image of repres
entations attached to CM elliptic curves (joint work with Benjamin York)\,
and we will discuss results on the minimal degree of definition of torsio
n structures (joint work with Enrique Gonzalez Jimenez).\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roy Zhao (University of California at Berkeley)
DTSTART:20221206T213000Z
DTEND:20221206T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/65
DESCRIPTION:Title: Heights on quaternionic Shimura varieties\nby Roy Zhao (Universi
ty of California at Berkeley) as part of MIT number theory seminar\n\nLect
ure held in Room 2-143 in the Simons Building (building 2).\n\nAbstract\nW
e give an explicit formula for the height of a special point on a quaterni
onic Shimura variety in terms of Faltings heights of CM abelian varieties.
This is a generalization of the work of Yuan and Zhang on proving the ave
raged Colmez conjecture. We also show an application of this formula to th
e Andre-Oort conjecture\, which was recently proven by Pila\, Shankar\, an
d Tsimerman.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicholas M. Katz and Pham Huu Tiep (Princeton\, Rutgers)
DTSTART:20230228T213000Z
DTEND:20230228T230000Z
DTSTAMP:20260404T094147Z
UID:MITNT/66
DESCRIPTION:Title: Local systems\, exponential sums\, and simple groups\nby Nichola
s M. Katz and Pham Huu Tiep (Princeton\, Rutgers) as part of MIT number th
eory seminar\n\nLecture held in Room 2-449 in the Simons Building (buildin
g 2).\n\nAbstract\nWe study the possible structure of monodromy groups of
Airy\, Kloosterman\, and hypergeometric $\\ell$-adic sheaves in characteri
stic $p$. We also discuss explicit constructions of local systems and thei
r related exponential sums on $\\mathbb{G}_m$ or $\\mathbb{A}^1$ that real
ize various (close to be) simple groups.\n\nThis will be a 2-part talk\, w
ith Katz giving part 1 starting at 4:35pm Eastern (via zoom\, projected on
the screen in 2-449)\, and Tiep giving part 2 starting at around 5:20pm i
n person in 2-449.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Li-Huerta (Harvard University)
DTSTART:20230418T203000Z
DTEND:20230418T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/67
DESCRIPTION:Title: On the plectic conjecture\nby Daniel Li-Huerta (Harvard Universi
ty) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in
the Simons Building (building 2).\n\nAbstract\nNekovář–Scholl observed
that the étale cohomology groups of Hilbert modular varieties enjoy the
action of a much larger profinite group than the absolute Galois group of
$\\mathbb{Q}$: the plectic Galois group. They conjectured that this
action extends to the level of complexes\, which would give a constructio
n of canonical classes in higher wedge powers of Selmer groups. I'll expla
in how this works\, as well as discuss analogues over local fields and glo
bal function fields.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Hamann (Princeton University)
DTSTART:20230404T203000Z
DTEND:20230404T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/68
DESCRIPTION:Title: Geometric Eisenstein series and the Fargues-Fontaine curve\nby L
inus Hamann (Princeton University) as part of MIT number theory seminar\n\
nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstra
ct\nGiven a connected reductive group $G$ and a Levi subgroup $M$\,\nBrave
rman-Gaitsgory and Laumon constructed geometric Eisenstein\nfunctors which
take Hecke eigensheaves on the moduli stack $\\operatorname{Bun}_{M}$ of\
n$M$-bundles on a smooth projective curve to eigensheaves on the moduli st
ack\n$\\operatorname{Bun}_{G}$ of\n$G$-bundles. Recently\, Fargues and Sch
olze constructed a general\ncandidate for the local Langlands corresponden
ce by doing geometric\nLanglands on the Fargues-Fontaine curve. In this ta
lk\, we explain recent work\non carrying the theory of geometric Eisenstei
n series over to the\nFargues-Scholze setting. In particular\, we explain
how\, given the\neigensheaf $S_{\\chi}$ on $\\operatorname{Bun}_{T}$ attac
hed to a smooth character $\\chi$ of\nthe maximal torus $T$\, one can cons
truct an eigensheaf on $\\operatorname{Bun}_{G}$ under\na certain generici
ty hypothesis on $\\chi$\, by applying a geometric\nEisenstein functor to
$S_{\\chi}$. Assuming the Fargues-Scholze\ncorrespondence satisfies certai
n expected properties\, we fully\ndescribe the stalks of this eigensheaf i
n terms of normalized\nparabolic inductions of the generic $\\chi$. This e
igensheaf has\nseveral useful applications to the study of the cohomology
of\nlocal and global Shimura varieties\, and time permitting we will\nexpl
ain such applications.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lie Qian (Stanford University)
DTSTART:20230411T203000Z
DTEND:20230411T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/69
DESCRIPTION:Title: Local Compatibility for Trianguline Representations\nby Lie Qian
(Stanford University) as part of MIT number theory seminar\n\nLecture hel
d in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nTriangul
ine representations are a big class of $p$-adic representations that conta
ins all nice enough (cristalline) ones but allow a continuous variation of
weights. Global consideration suggests that the $GL_2(\\mathbb{Q}_p)$ rep
resentation arising from a trianguline representation should have nonzero
eigenspace under Emerton's Jacquet functor. We prove this result using pur
ely local method as well as a generalization to $p$-adic representation of
$G_F$ for $F$ unramified over $\\mathbb{Q}_p$.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sameera Vemulapalli (Princeton University)
DTSTART:20230321T203000Z
DTEND:20230321T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/70
DESCRIPTION:Title: Counting low degree number fields with almost prescribed successive
minima\nby Sameera Vemulapalli (Princeton University) as part of MIT n
umber theory seminar\n\nLecture held in Room 2-449 in the Simons Building
(building 2).\n\nAbstract\nThe successive minima of an order in a degree $
n$ number field are $n$ real numbers encoding information about the Euclid
ean structure of the order. How many orders in degree n number fields are
there with almost prescribed successive minima\, fixed Galois group\, and
bounded discriminant? In this talk\, I will address this question for $n =
3\,4\,5$. The answers\, appropriately interpreted\, turn out to be piecew
ise linear functions on certain convex bodies. If time permits\, I will al
so discuss a geometric analogue of this problem: scrollar invariants of co
vers of $\\mathbb{P}^1$.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vadim Vologodsky (MIT)
DTSTART:20230307T213000Z
DTEND:20230307T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/71
DESCRIPTION:Title: Dual abelian varieties over a local field have equal volumes\nby
Vadim Vologodsky (MIT) as part of MIT number theory seminar\n\nLecture he
ld in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nA top d
egree differential form $\\omega$ on a smooth algebraic variety $X$ over a
local field $K$ gives rise to a (real valued) measure on $X(K)$. The Serr
e duality yields a natural isomorphism between the vector spaces of global
top degree forms on an abelian variety and the dual abelian variety. I wi
ll prove that the corresponding volumes are equal.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen D. Miller (Rutgers University)
DTSTART:20230509T203000Z
DTEND:20230509T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/72
DESCRIPTION:Title: The number theory and modular forms behind 8- and 24-dimensional sph
ere packing\nby Stephen D. Miller (Rutgers University) as part of MIT
number theory seminar\n\nLecture held in Room 2-449 in the Simons Building
(building 2).\n\nAbstract\nAlthough the solution to the sphere packing an
d "universal optimality" energy minimization problems in dimensions 8 and
24 have a very analytic flavor\, number theory is pervasive behind the sce
nes. I'll describe the rationality conjectures with Cohn which first poin
ted to the appearance of modular forms\, as well as Viazovska's interpolat
ion ansatz which more directly linked with modular forms\, especially for
energy minimization. (Joint work with Henry Cohn\, Abhinav Kumar\, Danylo
Radchenko\, and Maryna Viazovska.)\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amnon Besser (Ben-Gurion University/Boston University)
DTSTART:20230502T203000Z
DTEND:20230502T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/73
DESCRIPTION:Title: Local contributions to Quadratic Chabauty functions and derivatives
of Vologodsky functions with respect to $log(p)$\nby Amnon Besser (Ben
-Gurion University/Boston University) as part of MIT number theory seminar
\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbs
tract\nQuadratic Chabauty is a method for finding rational points on curve
s using $p$-adic methods. The quadratic Chabauty function is a function on
these rational points\, usually derived from some $p$-adic height\, which
is a sum of local terms at finite primes. The main term is the term at $p
$ which is a Coleman function\, but in order to make the method work one n
eeds to be able to compute the finite list of possible values of the other
contributions at primes of bad reduction.\n\nVologodsky functions are the
generalisation of Coleman functions to varieties with bad reduction. In t
his talk\, which is based on ongoing work with Steffen Muller and Padma Sr
inivasan\, I would like to promote the general (and vague) idea that the d
erivative of a Vologodsky integral with respect to the branch of log param
eter $log(p)$ is arithmetically interesting.\n\nAs an example I will show
how the local contribution above a prime $q$ to a $p$ adic height can be c
omputed by deriving the $q$-adic contribution to a $q$-adic height and use
this to obtain a computable formula for this contribution using the work
of Katz and Litt. In particular\, I will recover a formula of Betts and Do
gra for the local contribution to the Quadratic Chabauty function at a pri
me where the completion is a Mumford curve.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefano Marseglia (Utrecht University)
DTSTART:20230516T203000Z
DTEND:20230516T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/74
DESCRIPTION:Title: Cohen-Macaulay type of endomorphism rings of abelian varieties over
finite fields\nby Stefano Marseglia (Utrecht University) as part of MI
T number theory seminar\n\nLecture held in Room 2-449 in the Simons Buildi
ng (building 2).\n\nAbstract\nIn this talk\, we will speak about the (Cohe
n-Macaulay) type of the endomorphism ring of abelian varieties over a fini
te field with commutative endomorphism algebra. We will exhibit a conditio
n on the type of $\\mathrm{End}(A)$ implying that $A$ cannot be isomorphic
to its dual. In particular\, such an $A$ cannot be principally polarised
or a Jacobian. This is partly joint work with Caleb Springer.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Zhang (MIT)
DTSTART:20230912T203000Z
DTEND:20230912T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/75
DESCRIPTION:Title: Modular Gelfand pairs\, multiplicity-free triples\, and maybe some g
amma factors\nby Robin Zhang (MIT) as part of MIT number theory semina
r\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAb
stract\nThe classical theory of Gelfand pairs and its generalizations over
the complex numbers has many applications to number theory and automorphi
c forms\, such as the uniqueness of Whittaker models and the non-vanishing
of the central value of a triple product $L$-function. With an eye toward
s similar applications in the modular setting\, this talk presents an exte
nsion of the classical theory for representations of finite and compact gr
oups to such representations over algebraically closed fields with arbitra
ry characteristic. Time permitting\, I will also mention an analogue (join
t with J. Bakeberg\, M. Gerbelli-Gauthier\, H. Goodson\, A. Iyengar\, and
G. Moss) of the local converse theorem for Jacquet–Piatetski-Shapiro–S
halika gamma factors of mod $\\ell \\neq p$ representations of finite gene
ral linear groups.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Omer Offen (Brandeis University)
DTSTART:20230926T203000Z
DTEND:20230926T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/76
DESCRIPTION:Title: A new application of the residue method\nby Omer Offen (Brandeis
University) as part of MIT number theory seminar\n\nLecture held in Room
2-449 in the Simons Building (building 2).\n\nAbstract\nThe relative Langl
ands program studies the relations between functorial transfers of automor
phic representations from one group $G'$ to another group $G$\, period int
egrals over a subgroup $H$ of $G$ and special $L$-values.\nWhen $(G\,H)$ i
s a vanishing pair\, that is\, every cusp form of G has a vanishing $H$-pe
riod\, it is of interest to study discrete automorphic representations tha
t admit a non-vanishing $H$-period.\nFor this task\, Jacquet and Rallis de
veloped the residue method. \n It has since been used extensively\, mostly
for representations with cuspidal data lying in a maximal Levi subgroup o
f G. In this talk we focus on the case where $H=Sp(a)\\times Sp(b)$ lies i
n $G=Sp(a+b)$. We will introduce a new construction of some residual repre
sentations of G that admit H-periods. This is joint work in progress with
Sol Friedberg and David Ginzburg.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dubi Kelmer (Boston College)
DTSTART:20231024T203000Z
DTEND:20231024T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/77
DESCRIPTION:Title: Norm bounds on Eisenstein series\nby Dubi Kelmer (Boston College
) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in th
e Simons Building (building 2).\n\nAbstract\nIn this talk I will describe
some new results on the magnitude Eisenstein series corresponding to arith
metic lattices in hyperbolic space.\nAll new results are based on joint wo
rk with Shucheng Yu as well as with Alex Kontorovich and Cristopher Lutsk
o.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jerson Caro (Boston University)
DTSTART:20231003T203000Z
DTEND:20231003T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/78
DESCRIPTION:Title: A Chabauty-Coleman bound for surfaces\nby Jerson Caro (Boston Un
iversity) as part of MIT number theory seminar\n\nLecture held in Room 2-4
49 in the Simons Building (building 2).\n\nAbstract\nA celebrated result o
f Coleman gives a completely explicit version of Chabauty's finiteness the
orem for rational points in curves over a number field\, by a study of zer
os of p-adic analytic functions. \nAfter several developments around this
result\, the problem of proving an analogous explicit bound for higher dim
ensional subvarieties of abelian varieties remains elusive. In this talk\,
I'll sketch the proof of such a bound for surfaces contained in abelian v
arieties. This is a joint work with Hector Pasten.\n\nIn addition\, I'll p
resent an application of this method to give an upper bound for the number
of unexpected quadratic points of hyperelliptic curves of genus 3 defined
over $\\mathbb{Q}$. This is a joint work in progress with Jennifer Balakr
ishnan.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kestutis Cesnavicius (Université Paris-Saclay)
DTSTART:20231031T203000Z
DTEND:20231031T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/79
DESCRIPTION:Title: The affine Grassmannian as a presheaf quotient\nby Kestutis Cesn
avicius (Université Paris-Saclay) as part of MIT number theory seminar\n\
nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstra
ct\nThe affine Grassmannian of a reductive group $G$ is usually defined as
the étale sheafification of the quotient of the loop group $LG$ by the p
ositive loop subgroup. I will discuss various triviality results for $G$-t
orsors which imply that this sheafification is often not necessary.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spencer Leslie (Boston College)
DTSTART:20231107T213000Z
DTEND:20231107T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/80
DESCRIPTION:Title: Relative Langlands and Endoscopy\nby Spencer Leslie (Boston Coll
ege) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in
the Simons Building (building 2).\n\nAbstract\nIn this talk\, I will disc
uss the motivation for a theory of endoscopy in the context of the relativ
e Langlands program. I then outline the construction of endoscopic varieti
es\, which are spherical varieties of an associated endoscopic group. The
construction works for most hyperspherical varieties induced from symmetri
c varieties\, and relies on new rationality results for such varieties. We
highlight the role played by the dual Hamiltonian variety associated to a
symmetric variety a la Ben-Zvi\, Sakellaridis\, and Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rafael von Känel (IAS\, Tsinghua University)
DTSTART:20231121T210000Z
DTEND:20231121T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/81
DESCRIPTION:Title: Integral points on the Clebsch-Klein surfaces\nby Rafael von Kä
nel (IAS\, Tsinghua University) as part of MIT number theory seminar\n\nLe
cture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\
nIn this talk we present explicit bounds for the Weil height and the numbe
r of integral points on classical surfaces first studied by Clebsch (1871)
and Klein (1873). Building on Hirzebruch’s work in which he related the
se surfaces to a Hilbert modular surface\, we deduced our bounds from a ge
neral result for integral points on coarse Hilbert moduli schemes. After e
xplaining this deduction\, we discuss the strategy of proof of the general
result which combines the method of Faltings (Arakelov\,\nParsin\, Szpiro
) with modularity\, Masser-Wuestholz isogeny estimates\, and results based
on effective analytic estimates and/or Arakelov theory. Joint work with A
rno Kret.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hélène Esnault (FU Berlin/Harvard/Copenhagen)
DTSTART:20231205T210000Z
DTEND:20231205T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/83
DESCRIPTION:Title: Survey on some arithmetic properties of rigid local systems\nby
Hélène Esnault (FU Berlin/Harvard/Copenhagen) as part of MIT number theo
ry seminar\n\nLecture held in Room 2-449 in the Simons Building (building
2).\n\nAbstract\nA central conjecture of Simpson predicts that complex rig
id local systems on a smooth complex variety come from geometry. In the la
st couple of years\, we proved some arithmetic consequences of it: integra
lity (using the arithmetic Langlands program)\, F-isocrystal properties\,
crystallinity of the underlying p-adic representation (using the Cartier o
perator over the Witt vectors and the Higgs-de Rham flow) (for Shimura var
ieties of real rank at least 2\, this is the corner piece of Pila-Shankar-
Tsimerman's proof of the André-Oort conjecture)\, weak integrality of the
character variety (using de Jong's conjecture proved with the geometric L
anglands program) (yielding a new obstruction for a finitely presented gr
oup to be the topological fundamental group of a smooth complex variety).\
n\nWe'll survey some aspects of this (please ask if there is something on
which you would like me to focus on). The talk is based mostly on joint wo
rk with Michael Groechenig\, also\, even if less\, with Johan de Jong.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Sangiovanni Vincentelli (Princeton University)
DTSTART:20231212T210000Z
DTEND:20231212T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/84
DESCRIPTION:Title: Selmer groups\, p-adic L-functions and Euler Systems: A Unified Fram
ework.\nby Marco Sangiovanni Vincentelli (Princeton University) as par
t of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons
Building (building 2).\n\nAbstract\nSelmer groups are key invariants atta
ched to p-adic Galois representations. The Bloch—Kato conjecture predict
s a precise relationship between the size of certain Selmer groups and the
leading term of the L-function of the Galois representation under conside
ration. In particular\, when the L-function does not have a zero at s=0\,
it predicts that the Selmer group is finite and its order is controlled by
the value of the L-function at s=0. Historically\, one of the most powerf
ul tools to prove such relationships is by constructing an Euler System (E
S). \nAn Euler System is a collection of Galois cohomology classes over ra
mified abelian extensions of the base field that verify some co-restrictio
n compatibilities. The key feature of ESs is that they provide a way to bo
und Selmer groups\, thanks to the machinery developed by Rubin\, inspired
by earlier work of Thaine\, Kolyvagin\, and Kato. In this talk\, I will pr
esent joint work with C. Skinner\, in which we develop a new method for co
nstructing Euler Systems and apply it to build an ES for the Galois repres
entation attached to the symmetric square of an elliptic modular form. I w
ill stress how this method gives a unifying approach to constructing ESs\,
in that it can be successfully applied to retrieve most classical ESs (th
e cyclotomic units ES\, the elliptic units ES\, Kato’s ES…).\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordan Ellenberg (University of Wisconsin)
DTSTART:20240206T210000Z
DTEND:20240206T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/85
DESCRIPTION:Title: Around Smyth’s conjecture\nby Jordan Ellenberg (University of
Wisconsin) as part of MIT number theory seminar\n\nLecture held in Room 2-
449 in the Simons Building (building 2).\n\nAbstract\nAre there algebraic
numbers $x\,y\,z$ which are Galois conjugate to each other over $\\mathbb{
Q}$ and which satisfy the equation $5x + 6y + 7z = 0$? In 1986\, Chris Smy
th proposed an appealingly simple conjecture about linear relations betwee
n Galois conjugates\, which would provide answers to the above questions a
nd all questions of the same form\, and which has remained unsolved. My e
xperience is that most people\, upon seeing Smyth’s conjecture\, immedia
tely think it must be false (I certainly did!)\, but I have come to think
it’s true\, and I’ll talk about a provisional solution\, joint with Wi
ll Hardt\, in the case of three conjugates. I’ll explain why (as Smyth
observed) this is really a conjecture about linear combinations of permuta
tion matrices (related question\, solved by Speyer: which algebraic numbe
rs can be eigenvalues of the sum of two permutation matrices?)\, and why o
ur approach can be thought of as proving a “Hasse principle for probabil
ity distributions” in a particular case\, plus a bit of additive number
theory. Much of this talk\, maybe all\, will be suitable for undergraduat
es.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minhyong Kim (International Centre for Mathematical Sciences)
DTSTART:20240220T210000Z
DTEND:20240220T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/86
DESCRIPTION:Title: Arithmetic Quantum Field Theory?\nby Minhyong Kim (International
Centre for Mathematical Sciences) as part of MIT number theory seminar\n\
nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstra
ct\nMathematical structures suggested by quantum field theory have revolut
ionised important areas of algebraic geometry\, differential geometry\, as
well as topology in the last three decades. This talk will introduce a fe
w of the recent ideas for applying structures inspired by physics to arith
metic geometry.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ulrich Derenthal (Leibniz Universität Hannover)
DTSTART:20240227T210000Z
DTEND:20240227T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/87
DESCRIPTION:Title: Manin's conjecture for spherical Fano threefolds\nby Ulrich Dere
nthal (Leibniz Universität Hannover) as part of MIT number theory seminar
\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbs
tract\nWhen an algebraic variety over the rational numbers contains infini
tely many rational points\, we may study their distribution. In particular
\, for Fano varieties\, the asymptotic behavior of the number of rational
points of bounded height is predicted by Manin's conjecture.\n\nIn this ta
lk\, we discuss a proof of Manin's conjecture for smooth spherical Fano th
reefolds. In one case\, in order to obtain the expected asymptotic formula
\, it is necessary to exclude a thin subset with exceptionally many ration
al points from the count. This is joint work with V. Blomer\, J. Brüdern
and G. Gagliardi.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryan Chen (MIT)
DTSTART:20240305T210000Z
DTEND:20240305T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/88
DESCRIPTION:Title: Co-rank 1 Arithmetic Siegel–Weil\nby Ryan Chen (MIT) as part o
f MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Bu
ilding (building 2).\n\nAbstract\nI will introduce my recent work on an ar
ithmetic Siegel–Weil formula for Kudla–Rapoport $1$-cycles on integral
models of some unitary Shimura varieties. This formula implies that degre
es of Kudla–Rapoport arithmetic special $1$-cycles are encoded in the fi
rst derivatives of unitary Eisenstein series Fourier coefficients. In the
simplest case\, this can be rephrased in terms of Faltings heights of Heck
e translates of CM elliptic curves\, and the classical weight $2$ Eisenste
in series.\n\nThe key input is a new local limiting method which relates (
a) degrees of local special 0-cycles and (b) local contributions to height
s of special $1$-cycles.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura DeMarco (Harvard University)
DTSTART:20240312T203000Z
DTEND:20240312T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/89
DESCRIPTION:Title: From Manin–Mumford to dynamical rigidity\nby Laura DeMarco (Ha
rvard University) as part of MIT number theory seminar\n\nLecture held in
Room 2-449 in the Simons Building (building 2).\n\nAbstract\nIn the early
1980s\, Raynaud proved a theorem (the Manin–Mumford Conjecture) about th
e geometry of torsion points in abelian varieties\, using number-theoretic
methods. Around the same time\, and with completely different methods\,
McMullen proved a theorem about dynamical stability for maps on $\\mathbb{
P}^1$. In new work\, joint with Myrto Mavraki\, we view these results as
special cases of a unifying conjecture. The conjectural statement is dire
ctly inspired by a recent theorem of Gao and Habegger (called Relative Man
in–Mumford) and results in complex dynamics of Dujardin\, Gauthier\, Vig
ny\, and others.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Borys Kadets (Hebrew University)
DTSTART:20240402T203000Z
DTEND:20240402T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/90
DESCRIPTION:Title: Curves with many degree $d$ points\nby Borys Kadets (Hebrew Univ
ersity) as part of MIT number theory seminar\n\nLecture held in Room 2-449
in the Simons Building (building 2).\n\nAbstract\nWhen does a nice curve
$X$ over a number field $k$ have infinitely many closed points of degree $
d$?\nFaltings' theorem allows us to rephrase this problem in purely algebr
o-geometric terms\, though the resulting geometric question is far from be
ing fully solved. Previous work gave easy to state answers to the problem
for degrees $2$ (Harris-Silverman) and $3$ (Abramovich-Harris)\, but also
uncovered exotic constructions of such curves in all degrees $d \\geqslant
4$ (Debarre-Fahlaoui). I will describe recent progress on the problem\, w
hich answers the question in the large genus case. Along the way we uncove
r systematic explanations for the Debarre-Fahlaoui counstructions and prov
ide a complete geometric answer for $d \\leqslant 5$. The talk is based on
joint work with Isabel Vogt.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (Boston University)
DTSTART:20240416T203000Z
DTEND:20240416T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/92
DESCRIPTION:Title: A canonical algebraic cycle associated to a curve in its Jacobian\nby Padmavathi Srinivasan (Boston University) as part of MIT number theo
ry seminar\n\nLecture held in Room 2-449 in the Simons Building (building
2).\n\nAbstract\nThe Ceresa cycle is a canonical homologically trivial alg
ebraic cycle associated to a curve in its Jacobian. In his 1983 thesis\, C
eresa showed that this cycle is algebraically nontrivial for the generic c
urve over genus at least 3. Strategies for proving Fermat curves have infi
nite order Ceresa cycles due to B. Harris\, Bloch\, Bertolini-Darmon-Prasa
nna\, Eskandari-Murty use a variety of ideas ranging from computation of e
xplicit iterated period integrals\, special values of p-adic L functions a
nd points of infinite order on the Jacobian of Fermat curves. We will surv
ey many recent results around the Ceresa cycle\, and present ongoing work
with Jordan Ellenberg\, Adam Logan and Akshay Venkatesh where we produce m
any new explicit examples of curves over number fields with infinite order
Ceresa cycles.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent Pilloni (CNRS / Institut de Mathématiques d'Orsay)
DTSTART:20240430T203000Z
DTEND:20240430T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/93
DESCRIPTION:Title: On the modularity of abelian surfaces\nby Vincent Pilloni (CNRS
/ Institut de Mathématiques d'Orsay) as part of MIT number theory seminar
\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbs
tract\nWe prove that a positive proportion of abelian surfaces over the ra
tionals are modular. This is joint work with G. Boxer\, F. Calegari and T.
Gee.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tangli Ge (Princeton University)
DTSTART:20240507T203000Z
DTEND:20240507T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/94
DESCRIPTION:Title: Some height conjectures on abelian schemes\nby Tangli Ge (Prince
ton University) as part of MIT number theory seminar\n\nLecture held in Ro
om 2-449 in the Simons Building (building 2).\n\nAbstract\nMotivated by th
e conjectures of S. Zhang and Zilber–Pink\, I would like to formulate th
ree conjectures about height functions on abelian schemes. These conjectur
es will represent intersections of respectively unlikely\, just likely and
very likely kinds. The first one is a Bogomolov type conjecture. The seco
nd one is about boundedness of height. The third one is related to uniform
ity of the height bound. Some known results will be mentioned during the t
alk. I will then discuss an interesting implication: a specialization theo
rem for the Mordell–Weil group of an elliptic surface of Silverman.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Garcia (University College London)
DTSTART:20240917T203000Z
DTEND:20240917T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/96
DESCRIPTION:Title: The elliptic gamma function and Stark units\nby Luis Garcia (Uni
versity College London) as part of MIT number theory seminar\n\nLecture he
ld in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nI will
present a conjecture extending the classical theory of elliptic units from
imaginary quadratic fields to complex cubic fields. The role played by th
eta functions in the classical construction now corresponds to the ellipti
c gamma function\, a meromorphic function arising in mathematical physics.
Using this function we will define complex numbers that we conjecture to
lie on specified abelian extensions of cubic fields and to satisfy explici
t reciprocity laws. I will discuss some numerical and theoretical evidence
for these claims.\n\nThe talk will be based on arXiv:2311.04110 and is jo
int work with Nicolas Bergeron and Pierre Charollois.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Zubrilina (Massachusetts Institute of Technology)
DTSTART:20240924T203000Z
DTEND:20240924T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/97
DESCRIPTION:Title: Root number correlation bias of Fourier coefficients of modular form
s\nby Nina Zubrilina (Massachusetts Institute of Technology) as part o
f MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Bu
ilding (building 2).\n\nAbstract\nIn a recen study\, He\, Lee\, Oliver\, a
nd Pozdnyakov observed a striking oscillating pattern in the average value
of the P-th Frobenius trace of elliptic curves of prescribed rank and con
ductor in an interval range. Sutherland discovered that this bias extends
to Dirichlet coefficients of a much broader class of arithmetic L-function
s when split by root number. In my talk\, I will discuss this root number
correlation in families of holomorphic and Maass forms.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eran Assaf (Massachusetts Institute of Technology)
DTSTART:20241001T203000Z
DTEND:20241001T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/98
DESCRIPTION:Title: Hilbert modular forms from definite orthogonal modular forms\nby
Eran Assaf (Massachusetts Institute of Technology) as part of MIT number
theory seminar\n\nLecture held in Room 2-449 in the Simons Building (build
ing 2).\n\nAbstract\nIn this talk\, we explicitly determine the relationsh
ip between Hilbert modular forms and positive-definite orthogonal modular
forms\, with precise level structure and weight. This is achieved by analy
zing the interaction of the even Clifford functor with the p-neighbor rela
tion on lattices. \nBy connecting our results to the theory of theta corre
spondence\, we present an application to the non-vanishing of theta maps.\
nThis is joint work with Dan Fretwell\, Colin Ingalls\, Adam Logan\, Spenc
er Secord and John Voight.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Betts (Harvard University)
DTSTART:20241008T203000Z
DTEND:20241008T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/99
DESCRIPTION:Title: Lawrence--Venkatesh and the Section Conjecture\nby Alex Betts (H
arvard University) as part of MIT number theory seminar\n\nLecture held in
Room 2-449 in the Simons Building (building 2).\n\nAbstract\nGrothendieck
's famous Section Conjecture predicts that the set of rational points on a
smooth projective curve $X$ of genus at least two should be equal to a ce
rtain "section set" defined purely in terms of the etale fundamental group
of $X$. Despite several decades of interest\, this section set remains hi
ghly mysterious\, and we do not even know whether the section set is finit
e\, in accordance with the Mordell Conjecture.\n\nIn this talk I will desc
ribe work with Jakob Stix\, in which we applied the method of Lawrence--Ve
nkatesh to this question and proved a certain shadow of this expected fini
teness result.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Marcil (Columbia University)
DTSTART:20241022T203000Z
DTEND:20241022T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/100
DESCRIPTION:Title: $p$-adic $L$-functions for $P$-ordinary Hida families on unitary gr
oups\nby David Marcil (Columbia University) as part of MIT number theo
ry seminar\n\nLecture held in Room 2-449 in the Simons Building (building
2).\n\nAbstract\nI will first discuss the notion of automorphic representa
tions on a unitary group that are $P$-ordinary (at $p$)\, where $P$ is som
e parabolic subgroup. In the “ordinary” setting (i.e. when $P$ is mini
mal)\, such a representation $\\pi$ has a relatively simple structure at $
p$\, using a theorem of Hida. I will describe a generalization of the latt
er in the more general $P$-ordinary setting using the theory of types. I w
ill use this structure theorem to analyze and parametrize a $P$-ordinary H
ida family $C_\\pi$ associated to $\\pi$.\n\nThen\, I will introduce a $p$
-adic family of Eisenstein series that is “compatible” with $C_\\pi$.
Namely\, the Fourier coefficients of the former can be interpolated $p$-ad
ically to induce an “Eisenstein measure” and the family can be paired
with $C_\\pi$\, using an algebraic version of the doubling method\, to $p$
-adically interpolated special values of $L$-functions.\n\nI will conclude
by explaining how this Eisenstein measure corresponds to a $p$-adic $L$-f
unction for $C_\\pi$ viewed as an element of a $P$-ordinary Hecke algebra.
\n\nThese results generalize the ones obtained by Eischen-Harris-Li-Skinne
r in the ordinary setting and are from the speaker’s thesis.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cecilia Salgado (University of Groningen)
DTSTART:20241105T213000Z
DTEND:20241105T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/101
DESCRIPTION:Title: Mordell-Weil rank jumps on families of elliptic curves\nby Ceci
lia Salgado (University of Groningen) as part of MIT number theory seminar
\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbs
tract\nWe will present some recent developments around the variation of th
e Mordell-Weil rank in 1-dimensional families of elliptic curves\, by stud
ying them in the guise of elliptic surfaces. We will revisit Néron-Shioda
's construction of an infinite family of elliptic curves with rank at leas
t 11 and discuss ways of generalizing it to deal with certain elusive fami
lies.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Petrov (Massachusetts Institute of Technology)
DTSTART:20241112T213000Z
DTEND:20241112T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/102
DESCRIPTION:Title: Characteristic classes of p-adic local systems\nby Alexander Pe
trov (Massachusetts Institute of Technology) as part of MIT number theory
seminar\n\nLecture held in Room 2-449 in the Simons Building (building 2).
\n\nAbstract\nGiven an étale $\\mathbb{Z}_p$-local system of rank $n$ on
an algebraic variety $X$\, continuous cohomology classes of the group $GL_
n(\\mathbb{Z}_p)$ give rise to classes in (absolute) étale cohomology of
the variety. These characteristic classes can be thought of as p-adic anal
ogs of Chern-Simons characteristic classes of vector bundles with a flat c
onnection.\n\nOn a smooth projective variety over complex numbers\, Chern-
Simons classes of all flat bundles are torsion in degrees $>1$ by a theore
m of Reznikov. Likewise\, $p$-adic characteristic classes on smooth variet
ies over an algebraically closed field of characteristic zero vanish (at l
east for $p$ large as compared to the rank of the local system) in degrees
$>1$. But for varieties over non-closed fields the characteristic classes
of $p$-adic local systems turn out to often be non-zero even rationally.
When $X$ is defined over a $p$-adic field\, characteristic classes of a $p
$-adic local system on it can be partially expressed in terms of Hodge-the
oretic invariants of the local system. This relation is established throug
h considering an analog of Chern classes for vector bundles on the pro-ét
ale site of $X$.\n\nThis is joint work with Lue Pan.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kai-Wen Lan (University of Minnesota)
DTSTART:20241119T210000Z
DTEND:20241119T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/103
DESCRIPTION:Title: Some vanishing results for the rational completed cohomology of Shi
mura varieties\nby Kai-Wen Lan (University of Minnesota) as part of MI
T number theory seminar\n\nLecture held in Room 2-449 in the Simons Buildi
ng (building 2).\n\nAbstract\nI will start with some introduction to Shimu
ra varieties and their completed cohomology\, and report on my joint work
in progress with Lue Pan which shows that\, in the rational p-adic complet
ed cohomology of a general Shimura variety\, "sufficiently regular" infini
tesimal weights (whose meaning will be explained) can only show up in the
middle degree. I will give some examples and explain the main ingredients
in our work\, if time permits.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Bieker (Massachusetts Institute of Technology)
DTSTART:20241126T210000Z
DTEND:20241126T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/104
DESCRIPTION:Title: Heegner-Drinfeld cycles and a higher Gross-Zagier formula at deeper
level\nby Patrick Bieker (Massachusetts Institute of Technology) as p
art of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simo
ns Building (building 2).\n\nAbstract\nBy work of Yun-Zhang the self-inter
section number of Heegner-Drifeld cycles on moduli spaces of shtukas at Iw
ahori-level is related to (higher) derivatives of certain $L$-functions\,
providing a vast generalization of the Gross-Zagier formula in the functio
n field setting. \n\nIn this talk\, I will discuss integral models for cer
tain deeper level structures (like arbitrary\, i.e. possibly deeper than I
wahori\, $\\Gamma_0(\\Sigma)$-level) and explain how to construct Heegner-
Drinfeld cycles on them in order to formulate a generalization of the high
er GZ-formula.\n\nThis is partially based on joint work in progress with Z
hiwei Yun.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gyujin Oh (Columbia University)
DTSTART:20241217T210000Z
DTEND:20241217T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/106
DESCRIPTION:Title: Derived Hecke action for weight one modular forms via classicality<
/a>\nby Gyujin Oh (Columbia University) as part of MIT number theory semin
ar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nA
bstract\nIt is known that a p-adic family of modular forms does not necess
arily specialize into a classical modular form at weight one\, unlike the
modular forms of weight 2 or higher. We will explain how this "obstruction
to classicality" leads to a derived action on modular forms of weight one
\, which can also be understood as the so-called derived Hecke operator at
p. We will discuss the role of the derived action in the study of p-adic
periods of the adjoint of the weight one modular forms.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emre Can Sertöz (Leiden University)
DTSTART:20250113T210000Z
DTEND:20250113T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/107
DESCRIPTION:Title: Computing transcendence and linear relations of 1-periods\nby E
mre Can Sertöz (Leiden University) as part of MIT number theory seminar\n
\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstr
act\nI will sketch a modestly practical algorithm to compute all linear re
lations with algebraic coefficients between any given finite set of 1-peri
ods. As a special case\, we can algorithmically decide the transcendence o
f 1-periods. This is based on the "qualitative description" of these relat
ions by Huber and Wüstholz via 1-motives. We combine their result with th
e recent work on computing the endomorphism ring of abelian varieties. Thi
s is a work in progress with Jöel Ouaknine (MPI SWS) and James Worrell (O
xford).\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean Howe (University of Utah)
DTSTART:20250204T210000Z
DTEND:20250204T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/108
DESCRIPTION:Title: Sideways equidistribution of function field L-functions\nby Sea
n Howe (University of Utah) as part of MIT number theory seminar\n\nLectur
e held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nIn
the first part of this talk\, we will explain a concise description of the
asymptotic distributions of eigenvalues of Haar-random orthogonal matrice
s using a new $\\sigma$-moment generating function that replaces the usual
exponential with the plethystic exponential of symmetric function theory.
Similar descriptions can be obtained also for compact symplectic\, unitar
y\, and symmetric groups. \n\nIn the second part of the talk\, we will exp
lain how to use point-counting techniques to compute\, for a fixed finite
field $\\mathbb{F}_q$\, the distribution of the zeroes of the $L$-function
of a random smooth degree $d$ surface in $\\mathbb{P}^3_{\\mathbb{F}_q}$
as $d \\rightarrow \\infty$. The result is a simple description of the as
ymptotic $\\sigma$-moment generating function. Comparing this with our de
scription of the asymptotic distribution of the eigenvalues of a Haar-rand
om orthogonal matrix\, we obtain an equidistribution result that is "sidew
ays" compared to the equidistribution results obtained by Katz and Sarnak\
, i.e. where the order of the limits in $d$ and $q$ have been exchanged. T
his sideways equidistribution is finer in that it sees the stable cohomolo
gy of local systems in all degrees instead of just the zeroth degree neede
d to compute monodromy. \n\nThe techniques used are robust and apply also
to the L-functions of more general smooth hypersurface sections\, as well
as some simple Dirichlet characters that were previously studied by Bergst
röm-Diaconu-Petersen-Westerland. Time permitting\, we will briefly discus
s further generalizations and related work in progress joint with Bertucci
/ Bilu / Bilu and Das.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Mundy (Princeton University)
DTSTART:20250211T210000Z
DTEND:20250211T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/109
DESCRIPTION:Title: Vanishing of Selmer groups for Siegel modular forms\nby Samuel
Mundy (Princeton University) as part of MIT number theory seminar\n\nLectu
re held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nLe
t $\\pi$ be a cuspidal automorphic representation of $Sp_{2n}$ over $\\mat
hbb{Q}$ which is holomorphic discrete series at infinity\, and $\\chi$ a D
irichlet character. Then one can attach to $\\pi$ an orthogonal $p$-adic G
alois representation $\\rho$ of dimension $2n+1$. Assume $\\rho$ is irredu
cible\, that $\\pi$ is ordinary at $p$\, and that $p$ does not divide the
conductor of $\\chi$. I will describe work in progress which aims to prove
that the Bloch--Kato Selmer group attached to $\\rho\\otimes\\chi$ vanish
es\, under some mild ramification assumptions on $\\pi$\; this is what is
predicted by the Bloch--Kato conjectures.\n\nThe proof uses "ramified Eise
nstein congruences" by constructing $p$-adic families of Siegel cusp forms
degenerating to Klingen Eisenstein series of nonclassical weight\, and us
ing these families to construct ramified Galois cohomology classes for the
Tate dual of $\\rho\\otimes\\chi$.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Logan (Carleton University)
DTSTART:20250304T210000Z
DTEND:20250304T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/110
DESCRIPTION:Title: Kodaira dimension of Hilbert modular threefolds\nby Adam Logan
(Carleton University) as part of MIT number theory seminar\n\nLecture held
in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nFollowing
a method introduced by Thomas-Vasquez and developed by Grundman\,\nwe pro
ve that many Hilbert modular threefolds of arithmetic\ngenus $0$ and $1$ a
re of general type\, and that some are of nonnegative\nKodaira dimension.
The new ingredient is a detailed study\nof the geometry and combinatorics
of totally positive integral elements\n$x$ of a fractional ideal $I$ in a
totally real number field $K$ with\nthe property that tr $xy < $ min $I$
tr $y$ for some $y \\gg 0 \\in K$.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Stoll (Universität Bayreuth)
DTSTART:20250311T203000Z
DTEND:20250311T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/111
DESCRIPTION:Title: Conjectural asymptotics of prime orders of points on elliptic curv
es over number fields\nby Michael Stoll (Universität Bayreuth) as par
t of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons
Building (building 2).\n\nAbstract\nDefine\, for a positive integer $d$\,
$S(d)$ to be the set of all primes \n$p$ that occur as the order of a poi
nt $P \\in E(K)$ on an elliptic curve \n$E$ defined over a number field $K
$ of degree $d$. We discuss how some \nplausible conjectures on the sparsi
ty of newforms with certain \nproperties would allow us to deduce a fairly
precise result on the \nasymptotic behavior of $\\max S(d)$ as $d$ tends
to infinity.\n\nThis is joint work with Maarten Derickx.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rebecca Bellovin (University of Connecticut)
DTSTART:20250408T203000Z
DTEND:20250408T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/112
DESCRIPTION:Title: Characterizing perfectoid covers of abelian varieties\nby Rebec
ca Bellovin (University of Connecticut) as part of MIT number theory semin
ar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nA
bstract\nPerfectoid spaces have emerged as a key tool in p-adic Hodge theo
ry over the past decade\, generalizing earlier ideas due to people like Fo
ntaine and Wintenberger. I will discuss some history and applications of
this circle of ideas\, before talking about recent work characterizing per
fectoid covers of certain abelian varieties. This is joint work with Hanl
in Cai and Sean Howe.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vasily Dolgushev (Temple University)
DTSTART:20250415T203000Z
DTEND:20250415T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/113
DESCRIPTION:Title: The action of Grothendieck-Teichmueller (GT) shadows on child's dra
wings\nby Vasily Dolgushev (Temple University) as part of MIT number t
heory seminar\n\nLecture held in Room 2-449 in the Simons Building (buildi
ng 2).\n\nAbstract\nGrothendieck-Teichmueller (GT) shadows can be thought
of as approximations of elements of the mysterious Grothendieck-Teichmuell
er group GT introduced by V. Drinfeld in 1990. GT-shadows are morphisms of
a groupoid GTSh whose objects are certain finite index normal subgroups o
f the Artin braid group. The groupoid GTSh is closely connected to group G
T and to the absolute Galois group $G_Q$ of rational numbers. GTSh acts on
Grothendieck's child's drawings and this action is compatible with those
of the groups $G_Q$ and GT. In my talk\, I will present the hierarchy of o
rbits of child's drawings with respect to the actions of $G_Q$\, GT and GT
Sh\, give selected examples and say a few words about future directions of
this research. This talk is loosely based on my paper "The Action of GT-S
hadows on Child's Drawings" (J. of Algebra\, 2025). In many respects\, the
exploration of the action of GT-Shadows on child's drawings is inspired b
y \na paper written by D. Harbater and L. Schneps in 1997.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jit Wu Yap (Harvard University)
DTSTART:20250422T203000Z
DTEND:20250422T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/114
DESCRIPTION:Title: Quantitative Equidistribution of Small Points for Canonical Heights
\nby Jit Wu Yap (Harvard University) as part of MIT number theory semi
nar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\n
Abstract\nLet $K$ be a number field and $A$ an abelian variety over $K$. T
hen if $h_{\\operatorname{NT}}(x)$ denotes the Neron--Tate height of $x \\
in A(\\overline{\\mathbb{Q}})$\, Szpiro-Ullmo-Zhang showed that the Galois
orbits of a generic sequence $(x_n)$ with $h_{\\operatorname{NT}}(x_n) \\
to 0$ must equidistribute to the Haar measure of $A(\\mathbb{C})$. In this
talk\, I will explain a quantitative version of their equidistribution th
eorem along with its generalization to polarized dynamical systems.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia University)
DTSTART:20250429T190000Z
DTEND:20250429T200000Z
DTSTAMP:20260404T094147Z
UID:MITNT/115
DESCRIPTION:Title: Additive combinatorics and descent\nby Carlo Pagano (Concordia
University) as part of MIT number theory seminar\n\nLecture held in Room 2
-255 in the Simons Building (building 2).\n\nAbstract\nWe shall discuss th
e problem of constructing elliptic curves over number fields with positive
but "controlled" rank. An example of this problem is: given a quadratic e
xtension $L/K$ of number fields\, construct an elliptic curve $E/K$ such t
hat $0<\\text{rk}(E(K))=\\text{rk}(E(L))$. Another example is: for a numbe
r field $K$\, find an elliptic curve $E/K$ such that $\\text{rk}(E(K))=1$.
\nWith Peter Koymans we introduced a method to tackle this type of proble
ms\, combining additive combinatorics with 2-descent. I will explain our p
ast work on the former problem\, where we showed that Hilbert 10th problem
has negative answer on ring of integers of general number fields. Next\,
I will explain our joint work in progress\, where we settle the latter que
stion\, showing the following stronger result: if $E/K$ has full rational
$2$-torsion and no cyclic degree $4$ isogeny defined over $K$\, and it has
at least one quadratic twist with odd root number\, then it has infinitel
y many quadratic twists $d$ in $K^{\\ast}/K^{\\ast 2}$ such that $\\text{r
k}(E^d(K))=1$.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Park (Ohio State University)
DTSTART:20250513T190000Z
DTEND:20250513T200000Z
DTSTAMP:20260404T094147Z
UID:MITNT/116
DESCRIPTION:Title: The Quadratic Manin-Peyre conjecture for del Pezzo surfaces\nby
Jennifer Park (Ohio State University) as part of MIT number theory semina
r\n\nLecture held in Room 4-149 in the Mclaurin Buildings (building 4).\n\
nAbstract\nManin-Peyre conjecture\, counting point of bounded height on Fa
no varieties\, has been the subject of intense research in the past few de
cades. We provide a general framework for the Manin-Peyre conjecture for t
he symmetric square of any del Pezzo surface X\, and prove the conjecture
for the infinite family of nonsplit quadric surfaces. Previously\, there w
ere only two examples in the literature: Sym^2(P^2) and Sym^2(P^1 x P^1).
In order to achieve the predicted asymptotic\, we show that a type II thi
n set of a new flavour must be removed. A key tool we develop and that can
be applied to further examples is a result for summing multiplicative fun
ctions and Euler products over quadratic extensions. To establish our coun
ting result for the specific family of quadric surfaces\, we improve exist
ing lattice point counting results in the literature and make crucial use
of a novel form of lattice point counting. This work is joint with Frances
ca Balestrieri\, Kevin Destagnol\, Julian Lyczak\, and Nick Rome.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sho Tanimoto (Nagoya University)
DTSTART:20250930T203000Z
DTEND:20250930T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/117
DESCRIPTION:Title: Homological sieve and Manin's conjecture\nby Sho Tanimoto (Nago
ya University) as part of MIT number theory seminar\n\nLecture held in Roo
m 2-449 in the Simons Building (building 2).\n\nAbstract\nI present our pr
oofs for a version of Manin's conjecture over $\\mathbb F_q$ for $q$ large
and Cohen—Jones—Segal conjecture over $\\mathbb C$ for rational curve
s on split quartic del Pezzo surfaces. The proofs share a common method wh
ich builds upon prior work of Das—Tosteson. We call this method as homol
ogical sieve method. The main ingredients of this method are (i) the const
ruction of bar complexes formalizing the inclusion-exclusion principle and
its point counting estimates\, (ii) dimension estimates for spaces of rat
ional curves using conic bundle structures\, (iii) estimates of error term
s using arguments of Sawin—Shusterman based on Katz's results\, and (iv)
a certain virtual height zeta function revealing the compatibility of bar
complexes and Peyre's constant. Our argument verifies the heuristic appro
ach to Manin's conjecture over global function fields given by Batyrev and
Ellenberg-Venkatesh\, and it is a nice combination of various tools from
algebraic geometry (birational geometry of moduli spaces of rational curve
s)\, arithmetic geometry (simplicial schemes\, their homotopy theory\, and
Grothendieck—Lefschetz trace formula)\, algebraic topology (the inclusi
on-exclusion principle and Vassiliev type method of the bar complexes) and
some elementary analytic number theory. This is joint work with Ronno Das
\, Brian Lehmann\, and Phil Tosteson with a help by Will Sawin and Mark Sh
usterman.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sachi Hashimoto (Brown University)
DTSTART:20250909T203000Z
DTEND:20250909T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/118
DESCRIPTION:Title: Rational points on $X_0(N)^*$ when $N$ is non-squarefree\nby Sa
chi Hashimoto (Brown University) as part of MIT number theory seminar\n\nL
ecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract
\nThe rational points of the modular curve $X_0(N)$ classify pairs $(E\,C_
N)$ of elliptic curves over $\\mathbb{Q}$ together with a rational cyclic
subgroup of order $N$. The curve $X_0(N)^*$ is the quotient of $X_0(N)$ by
the full group of Atkin-Lehner involutions. Elkies showed that the ration
al points on this curve classify elliptic curves over the algebraic closur
e of $\\mathbb{Q}$ that are isogenous to their Galois conjugates\, and con
jectured that when $N$ is large enough\, the points are all CM or cuspidal
. In joint work with Timo Keller and Samuel Le Fourn\, we study the ration
al points on the family $X_0(N)^*$ for N non-squarefree. In particular we
will report on some integrality results for the $j$-invariants of points o
n $X_0(N)^*$.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikayel Mkrtchyan (MIT)
DTSTART:20250916T203000Z
DTEND:20250916T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/119
DESCRIPTION:Title: Higher Siegel-Weil formula for unitary groups over function fields:
case of corank-1 coefficients\nby Mikayel Mkrtchyan (MIT) as part of
MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Buil
ding (building 2).\n\nAbstract\nThe arithmetic Siegel-Weil formula relates
degrees of special cycles on Shimura varieties to derivatives of certain
Eisenstein series. In their seminal work\, Feng-Yun-Zhang have defined ana
logous special cycles on moduli spaces of shtukas over function fields\, a
nd proved a higher Siegel-Weil formula relating degrees of special cycles
on moduli spaces of shtukas with r legs\, to r-th derivatives of non-degen
erate Fourier coefficients of the Eisenstein series. In this talk\, I will
report on joint work with Tony Feng and Benjamin Howard\, where we prove
a higher Siegel-Weil formula for corank-1 singular Fourier coefficients. A
key feature of the proof is an unexpected full support property of the re
levant "Hitchin" fibration.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Achter (Colorado State University)
DTSTART:20250923T203000Z
DTEND:20250923T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/120
DESCRIPTION:Title: Torsion finite problems\nby Jeff Achter (Colorado State Univers
ity) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in
the Simons Building (building 2).\n\nAbstract\nConsider an abelian variet
y A over a number field K. The torsion\nsubgroup of A(K) is finite\; a re
sult of Ribet shows that this finiteness\npersists over the cyclotomic ext
ension of K.\n\nNow consider a second abelian variety B/K\, and the infini
te extension\nK_B generated by the coordinates of its torsion points. Con
ditional\non the Mumford-Tate conjecture (and up to a finite extension of
K)\,\nI will give a criterion for the finitude of the torsion subgroup of\
nA(K_B). I'll also describe a motivic generalization of\nthis story\, whi
ch in retrospect explains certain\nalgebraic cycles we discovered on tors
ion-infinite pairs of CM abelian\nvarieties. (Joint work with Lian Duan an
d Xiyuan Wang.)\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Harris Daniels (Amherst College)
DTSTART:20251007T203000Z
DTEND:20251007T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/121
DESCRIPTION:Title: Near coincidences and nilpotent division fields of elliptic curves<
/a>\nby Harris Daniels (Amherst College) as part of MIT number theory semi
nar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\n
Abstract\nA natural question about the division fields of a fixed elliptic
curve $E/\\mathbb{Q}$ is whether there is a coincidence between the divis
ion fields. I.e. Are there distinct integers $m \\neq n$ such that the $m
$-division field equals the $n$-division field. In 2023\, Daniels and Loza
no-Robledo gave partial answers to this question\, using (among other tool
s) the fact that the $n$-th roots of unity often fail to lie in the $m$-di
vision field\, thereby preventing such coincidences.\n\nMotivated by this\
, we consider a broader notion of \\emph{near coincidences}: when does the
re exist $E/\\mathbb{Q}$ and distinct $m\,n$ such that\n\\[\n\\mathbb{Q}(E
[n]) = \\mathbb{Q}(E[m]\, \\zeta_n)?\n\\]\nIn the first part of this talk\
, we answer this question completely in the case where $m$ and $n$ are pow
ers of the same prime.\n\nIn the second part\, we turn to a seemingly unre
lated but natural problem: classifying all elliptic curves $E/\\mathbb{Q}$
and positive integers $n$ such that\n\\[\n\\operatorname{Gal}(\\mathbb{Q}
(E[n])/\\mathbb{Q})\n\\]\nis a nilpotent group. This question generalizes
the classification of abelian division fields obtained by Gonz\\'alez-Jim\
\'enez and Lozano-Robledo (2016). We present a conditionally complete clas
sification of nilpotent division fields\, under either a standard conjectu
re about rational points on modular curves attached to normalizers of non-
split Cartan subgroups or a full classification of the Mersenne primes. Th
is is joint work with Jeremy Rouse.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uriya First (University of Haifa)
DTSTART:20251028T203000Z
DTEND:20251028T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/123
DESCRIPTION:Title: Higher Essential Dimension: First Steps\nby Uriya First (Univer
sity of Haifa) as part of MIT number theory seminar\n\nLecture held in Roo
m 2-449 in the Simons Building (building 2).\n\nAbstract\nLet $G$ be a lin
ear algebraic group over a field $k$.\nLoosely speaking\, the essential di
mension of $G$ measures\nthe number of independent parameters that are req
uired to define\na $G$-torsor over a $k$-field. It measures the complexity
of $G$-torsors and equivalent objects.\nOne formal way to define it is to
say\nthat the essential dimension of $G$ is $\\leq m$ if every $G$-torsor
over a finite-type $k$-scheme is\, away from some codimension-$1$ closed
subscheme\, the specialization of a $G$-torsor over a finite-type $k$-sche
me of dimension $m$.\n\nRecently\, for every integer $d\\geq 0$\, we\ndefi
ned the $d$-essential dimension of $G$\,\ndenoted $\\mathrm{ed}^{(d)}(G)$\
, by replacing ``codimension-$1$'' with ``codimension-$(d+1)$''.\nAfter re
calling ordinary essential dimension and its usages\, I will discuss work
in progress about the new sequence of invariants $\\{\\mathrm{ed}^{(d)}(G)
\\}_{d\\geq 0}$ and its asymptotic behavior as $d\\to \\infty$.\nFor examp
le\, $\\mathrm{ed}^{(d)}(\\mathbf{G}_m)=d$\, $\\mathrm{ed}^{(d)}(\\mathbf{
\\mu}_n)=d+1$ and\n$\\mathrm{ed}^{(d)}(\\mathbf{G}_m\\times\\mathbf{G}_m)=
2d$ in characteristic $0$. Moreover\, there is a dichotomy between unipote
nt and non-unipotent\ngroups: If $G$ is unipotent\, the sequence $\\{\\mat
hrm{ed}^{(d)}(G)\\}_{d\\geq 0}$ is bounded\,\nwhereas if $G$ is not unipo
tent\, then $\\mathrm{ed}^{(d)}(G)\\geq d-C_G$ for some constant $C_G$.\nT
here are also some interesting anomalies.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jennifer Paulhus (Mount Holyoke College)
DTSTART:20251104T213000Z
DTEND:20251104T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/124
DESCRIPTION:Title: Automorphism Groups of Riemann Surfaces\nby Jennifer Paulhus (M
ount Holyoke College) as part of MIT number theory seminar\n\nLecture held
in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nClassific
ation questions about automorphisms of compact Riemann surfaces date back
to the 1800s. There has been renewed interest in these questions over the
last 30 years as advances in computation have provided new ways to explore
the area. We will talk about some of those advancements focusing on group
s which are automorphisms in just about every genus they should be (partic
ularly simple groups and the alternating groups $A_n$).\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Princeton University)
DTSTART:20251125T210000Z
DTEND:20251125T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/125
DESCRIPTION:Title: Arithmetic of Fourier coefficients of Gan-Gurevich lifts on $\\math
sf{G}_2$\nby Naomi Sweeting (Princeton University) as part of MIT numb
er theory seminar\n\nLecture held in Room 2-449 in the Simons Building (bu
ilding 2).\n\nAbstract\nQuaternionic modular forms on $\\mathsf{G}_2$ carr
y a surprisingly rich arithmetic structure. For example\, they have a theo
ry of Fourier expansions where the Fourier coefficients are indexed by tot
ally real cubic rings. For quaternionic modular forms on $\\mathsf{G}_2$ a
ssociated via functoriality with certain modular forms on $\\mathrm{PGL}_2
$\, Gross conjectured in 2000 that their Fourier coefficients encode $L$-v
alues of cubic twists of the modular form (echoing Waldspurger's work on F
ourier coefficients of half-integral weight modular forms). This talk will
report on recent work proving Gross's conjecture when the modular forms a
re dihedral\, giving the first examples for which it is known. Based on jo
int work with Petar Bakic\, Alex Horawa\, and Siyan Daniel Li-Huerta.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Roberts (University of Minnesota\, Morris)
DTSTART:20251014T203000Z
DTEND:20251014T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/126
DESCRIPTION:Title: Wild Ramification in Hypergeometric Motives\nby David Roberts (
University of Minnesota\, Morris) as part of MIT number theory seminar\n\n
Lecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstrac
t\nThe bulk of my talk will be an overview of the current state of knowled
ge of wild ramification in general hypergeometric motives at a fixed prime
$p$. The presentation will be as elementary and visual as possible\, usi
ng p-adic ordinals of field discriminants of trinomials $x^n - n t x + (n
-1) t$ and their underlying Galois theory as a continuing example. It wil
l be revealed that the general situation is very complicated\, but exhibit
s enough patterns that one can still reasonably hope for a universal formu
la identifying all numerical invariants of wild p-adic ramification in all
hypergeometric motives.\n\nIf one restricts to the case where $\\operator
name{ord}_p(t)$ is coprime to $p$ then the situation simplifies considerab
ly. The ramp conjecture of Section 13 of my survey on Hypergeometric Moti
ves with Fernando Rodriguez Villegas predicts conductor exponents. I wil
l conclude with a new refinement of the ramp conjecture that predicts\, vi
a Feynman-like diagrams\, how the conductor exponents decompose as a sum o
f slopes. The refinement reveals much more structure than the original r
amp conjecture\, and I hope will point the way to a proof.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (Amherst College)
DTSTART:20260217T210000Z
DTEND:20260217T220000Z
DTSTAMP:20260404T094147Z
UID:MITNT/127
DESCRIPTION:Title: Angle ranks of Abelian varieties\nby David Zureick-Brown (Amher
st College) as part of MIT number theory seminar\n\nLecture held in Room 2
-449 in the Simons Building (building 2).\n\nAbstract\nI will discuss an e
lementary notion -- the rank of the multiplicative group generated by root
s of a polynomial. For Weil polynomials\, the roots lie on a circle and on
e calls this the angle rank. I'll present new results about angle ranks a
nd give some applications to the Tate conjecture for Abelian varieties ove
r finite fields and to arithmetic statistics.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Duker Lichtman (Stanford and Math\, Inc) and Jesse Han (Math
\, Inc.)
DTSTART:20250918T183000Z
DTEND:20250918T193000Z
DTSTAMP:20260404T094147Z
UID:MITNT/128
DESCRIPTION:Title: Gauss - an agentic formalization of the Prime Number Theorem\nb
y Jared Duker Lichtman (Stanford and Math\, Inc) and Jesse Han (Math\, Inc
.) as part of MIT number theory seminar\n\nLecture held in Room 2-190 in t
he Simons Building (building 2).\n\nAbstract\nIn this talk we'll highlight
some recent formalization advances using a\nnew agent\, Gauss. In particu
lar\, with Gauss we obtained a Lean proof of\nthe Prime Number Theorem in
strong form\, completing a challenge set in\nJanuary 2024 by Alex Kontorov
ich and Terry Tao. We hope Gauss will help\nassist working mathematicians\
, especially those who do not write formal\ncode themselves.\n\nNote the u
nusual day/time/place!\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gaëtan Chenevier (ENS PSL)
DTSTART:20260224T180000Z
DTEND:20260224T190000Z
DTSTAMP:20260404T094147Z
UID:MITNT/129
DESCRIPTION:Title: Unimodular lattices of rank 29 and applications\nby Gaëtan Che
nevier (ENS PSL) as part of MIT number theory seminar\n\nLecture held in R
oom 2-255 in the Simons Building (building 2).\n\nAbstract\nI will explain
the recent classification of rank 29 unimodular Euclidean integral lattic
es and discuss some applications to modular forms for GL_n(Z). Joint work
with Olivier Taïbi.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Piotr Achinger (IMPAN Warsaw and Kyiv School of Economics)
DTSTART:20260303T213000Z
DTEND:20260303T223000Z
DTSTAMP:20260404T094147Z
UID:MITNT/130
DESCRIPTION:Title: Gluing triples\nby Piotr Achinger (IMPAN Warsaw and Kyiv School
of Economics) as part of MIT number theory seminar\n\nLecture held in Roo
m 2-449 in the Simons Building (building 2).\n\nAbstract\nFor a scheme ove
r $\\mathbb{Q}_p$\, we wish to describe the ways of extending over $\\math
bb{Z}_p$ using formal and rigid geometry. Our main theorem does this over
arbitrary (excellent) base provided we replace schemes with separated alge
braic spaces. This result can be seen as a version of Beauville–Laszlo/A
rtin gluing of coherent sheaves\, but for spaces\, and turns out to be clo
sely related to Artin's contraction theorem. This is joint work with Alex
Youcis.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Pappas (Michigan State University)
DTSTART:20260331T190000Z
DTEND:20260331T200000Z
DTSTAMP:20260404T094147Z
UID:MITNT/133
DESCRIPTION:Title: Toric schemes and integral models for Shimura varieties\nby Geo
rge Pappas (Michigan State University) as part of MIT number theory semina
r\n\nLecture held in Room 4-231 at MIT (Building 4).\n\nAbstract\nI will e
xplain a conjectural description of the local structure of $p$-adic integr
al models for Shimura varieties with level structures "of type $\\Gamma_1
(p)$”. This aims to generalize a classical result of Deligne-Rapoport ab
out the integral model of the modular curve $X_1(p)$. The description invo
lves two ingredients: A toric scheme which is constructed directly from th
e local Shimura datum and which is related to the local model of a Shimura
variety for parahoric level\, and a Galois cover which is a toric extensi
on of the Lang map. This is joint work with M. Rapoport.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Haoyang Guo (University of Minnesota)
DTSTART:20260407T203000Z
DTEND:20260407T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/134
DESCRIPTION:Title: A primitive purity theorem for Frobenius modules\nby Haoyang Gu
o (University of Minnesota) as part of MIT number theory seminar\n\nLectur
e held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nIn
p-adic Hodge theory\, a fundamental observation of Breuil and Kisin is tha
t some Galois representations over p-adic integers give rise to interestin
g integral linear-algebraic data\, where the latter nowadays are called Br
euil--Kisin modules. The notion naturally generalizes to modules with Frob
enius structures over a more general base\, thanks to the prismatic cohomo
logy introduced by Bhatt and Scholze. In this talk\, we show that Frobeniu
s modules over a regular ring admit a primitive purity theorem\, and expla
in its potential application to p-adic local systems.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Weinstein (Boston University)
DTSTART:20260421T203000Z
DTEND:20260421T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/135
DESCRIPTION:by Jared Weinstein (Boston University) as part of MIT number t
heory seminar\n\nLecture held in Room 2-449 in the Simons Building (buildi
ng 2).\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qiao He (Columbia University)
DTSTART:20260428T190000Z
DTEND:20260428T200000Z
DTSTAMP:20260404T094147Z
UID:MITNT/136
DESCRIPTION:by Qiao He (Columbia University) as part of MIT number theory
seminar\n\nLecture held in Room 4-231 at MIT (Building 4).\nAbstract: TBA\
n
LOCATION:https://stable.researchseminars.org/talk/MITNT/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Shusterman (Weizmann Institute of Science)
DTSTART:20260505T203000Z
DTEND:20260505T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/137
DESCRIPTION:by Mark Shusterman (Weizmann Institute of Science) as part of
MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Buil
ding (building 2).\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Urbanik (IAS)
DTSTART:20260512T203000Z
DTEND:20260512T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/138
DESCRIPTION:by David Urbanik (IAS) as part of MIT number theory seminar\n\
nLecture held in Room 2-449 in the Simons Building (building 2).\nAbstract
: TBA\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristian Popescu (UC San Diego & IAS Princeton)
DTSTART:20260331T203000Z
DTEND:20260331T213000Z
DTSTAMP:20260404T094147Z
UID:MITNT/139
DESCRIPTION:Title: An equivariant Tamagawa number formula for abelian $t$-motives and
applications\nby Cristian Popescu (UC San Diego & IAS Princeton) as pa
rt of MIT number theory seminar\n\nLecture held in Room 4-231 at MIT (Buil
ding 4).\n\nAbstract\nWe will explain the construction of a Galois equivar
iant Goss-type $L$-function associated to an abelian $t$-motive and outlin
e the formulation and proof of a Tamagawa Number Formula for its special v
alues at positive integers. This generalizes to the abelian $t$-motive and
equivariant settings Taelman's celebrated class-number formula for Drinfe
ld modules. If time permits\, we will show how the main result implies ana
logues of the Brumer-Stark and Coates-Sinnott Conjectures for abelian $t$
-motives. This is based on joint work with Ferrara\, Green\, Higgins and R
amachandran.\n
LOCATION:https://stable.researchseminars.org/talk/MITNT/139/
END:VEVENT
END:VCALENDAR