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BEGIN:VEVENT
SUMMARY:Romyar Sharifi (UCLA)
DTSTART:20200413T230000Z
DTEND:20200413T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/1
DESCRIPTION:Title: Eisenstein cocycles in motivic cohomology\nby Romyar Sharifi (
UCLA) as part of UCLA Number Theory Seminar\n\n\nAbstract\nI will describe
joint work with Akshay Venkatesh on the construction of GL_2(Z)-cocycles
valued in second K-groups of the function fields of the squares of the mul
tiplicative group over the rationals and a universal elliptic curve. Focu
sing on the first case\, I’ll explain how the cocycle we construct speci
alizes to homomorphisms taking modular symbols for congruence subgroups to
special elements in second cohomology groups of cyclotomic fields and sat
isfies a certain property of being “Eisenstein”.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jayce Getz (Duke)
DTSTART:20200427T230000Z
DTEND:20200427T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/2
DESCRIPTION:Title: On triple product L-functions\nby Jayce Getz (Duke) as part of
UCLA Number Theory Seminar\n\n\nAbstract\nEstablishing the conjectured an
alytic properties of triple product L-functions is a crucial case of Langl
ands functoriality. However\, little is known. I will present work in prog
ress on the case of triples of automorphic representations on GL_3\; in so
me sense this is the smallest case that appears out of reach via standard
techniques. The approach involves a relative trace formula and Poisson sum
mation on spherical varieties in the sense of Braverman-Kazhdan\, Ngo\, an
d Sakellaridis.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zheng Liu (UC Santa Barbara)
DTSTART:20200518T230000Z
DTEND:20200518T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/3
DESCRIPTION:Title: The doubling archimedean zeta integrals for unitary groups\nby
Zheng Liu (UC Santa Barbara) as part of UCLA Number Theory Seminar\n\n\nA
bstract\nIn order to verify the compatibility between the conjecture of Co
ates--Perrin-Riou and the interpolation results of the $p$-adic $L$-functi
ons constructed by using the doubling method\, a doubling archimedean zeta
integral needs to be calculated for holomorphic discrete series. When the
holomorphic discrete series is of scalar weight\, it has been done by Boc
herer-Schmidt and Shimura. In this talk\, I will explain a way to compute
this archimedean zeta integral for unitary groups of arbitrary signatures
and general vector weights. This is a joint work with Ellen Eischen.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao (Harvard)
DTSTART:20200527T230000Z
DTEND:20200527T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/4
DESCRIPTION:Title: Crystalline comparison of $\\mathrm{A}_\\mathrm{inf}$-cohomology\nby Zijian Yao (Harvard) as part of UCLA Number Theory Seminar\n\n\nAbs
tract\nA major goal of $p$-adic Hodge theory is to relate arithmetic struc
tures coming from various cohomology theories of $p$-adic varieties. Such
comparisons are usually achieved by constructing intermediate cohomology t
heories. A somewhat recent successful theory\, namely the $\\mathrm{A}_\\m
athrm{inf}$-cohomology\, has been invented by Bhatt--Morrow--Scholze\, ori
ginally via perfectoid spaces. In this talk\, I will describe a simpler ap
proach to prove the comparison between $\\mathrm{A}_\\mathrm{inf}$-cohomol
ogy and the crystalline cohomology over Fontaine's period ring $\\mathrm{A
}_\\mathrm{cris}$\, using flat descent of cotangent complexes. This approa
ch also allows us to prove compatibilities of certain $p$-adic filtrations
. Time permitting\, I will discuss some work in progress (partially joint
with Hansheng Diao) in the semistable/logarithmic case.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Stubley (UChicago)
DTSTART:20200601T230000Z
DTEND:20200601T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/5
DESCRIPTION:Title: Locally Split Galois Representations and Hilbert Modular Forms of
Partial Weight One\nby Eric Stubley (UChicago) as part of UCLA Number
Theory Seminar\n\n\nAbstract\nThe $p$-adic Galois representation attached
to a $p$-ordinary eigenform is upper triangular when restricted to a decom
position group at $p$. A natural question to ask is under what conditions
this upper triangular decomposition splits as a direct sum. Ghate and Vats
al have shown that for the Galois representation attached to a Hida family
of $p$-ordinary eigenforms\, the restriction to a decomposition group at
$p$ is split if and only if the family has complex multiplication\; in the
ir proof\, the weight one members of the family play a key role.\n\nI'll t
alk about work in progress which aims to answer similar questions in the c
ase of Galois representations for a totally real field which are split at
only some of the decomposition groups at primes above $p$. In this work Hi
lbert modular forms of partial weight one play a central role\; I'll discu
ss what is known about them and to what extent the techniques of Ghate and
Vatsal can be adapted to this situation.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (University of Cambridge)
DTSTART:20200420T220000Z
DTEND:20200420T225000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/6
DESCRIPTION:Title: Symmetric power functoriality for holomorphic modular forms\nb
y Jack Thorne (University of Cambridge) as part of UCLA Number Theory Semi
nar\n\n\nAbstract\nLanglands’s functoriality conjectures predict the exi
stence of "liftings" of automorphic representations along morphisms of L-g
roups. A basic case of interest comes from the irreducible algebraic repre
sentations of GL(2)\, thought of as the L-group of the reductive group GL(
2) over Q. I will discuss the proof\, joint with James Newton\, of the ex
istence of the corresponding functorial liftings for a broad class of holo
morphic modular forms\, including Ramanujan’s Delta function.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bas Edixhoven (Universiteit Leiden)
DTSTART:20200504T180000Z
DTEND:20200504T185000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/7
DESCRIPTION:Title: Geometric quadratic Chabauty\nby Bas Edixhoven (Universiteit L
eiden) as part of UCLA Number Theory Seminar\n\n\nAbstract\nJoint work wit
h Guido Lido (see arxiv preprint). Determining all rational points on a cu
rve of genus at least 2 can be difficult. Chabauty's method (1941) is to i
ntersect\, for a prime number p\, in the p-adic Lie group of p-adic points
of the jacobian\, the closure of the Mordell-Weil group with the p-adic p
oints of the curve. If the Mordell-Weil rank is less than the genus then t
his method has never failed. Minhyong Kim's non-abelian Chabauty programme
aims to remove the condition on the rank. The simplest case\, called quad
ratic Chabauty\, was developed by Balakrishnan\, Dogra\, Mueller\, Tuitman
and Vonk\, and applied in a tour de force to the so-called cursed curve (
rank and genus both 3). Our work aims to make the quadratic Chabauty metho
d small and geometric again\, by describing it in terms of only "simple al
gebraic geometry" (line bundles over the jacobian and models over the inte
gers).\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michele Fornea (Princeton)
DTSTART:20200511T230000Z
DTEND:20200511T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/8
DESCRIPTION:Title: Hirzebruch-Zagier classes and rational elliptic curves over quinti
c fields\nby Michele Fornea (Princeton) as part of UCLA Number Theory
Seminar\n\n\nAbstract\nIn joint work with Zhaorong Jin\, we establish new
instances of the BSD-conjecture for rational elliptic curves over quintic
fields. \nThe proof is p-adic in nature and relies on the comparison of tw
o rigid analytic functions: the automorphic p-adic L-function retaining in
formation about special L-values\, and the motivic p-adic L-function arisi
ng from cycles on Shimura threefolds. The construction of the latter funct
ion is of independent interest as it could be applied to other settings re
lated to the Gan-Gross-Prasad conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bao Le Hung (Northwestern)
DTSTART:20200928T230000Z
DTEND:20200928T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/9
DESCRIPTION:Title: Moduli of Fontaine-Laffaille modules and mod $p$ local-global comp
atibility\nby Bao Le Hung (Northwestern) as part of UCLA Number Theory
Seminar\n\n\nAbstract\nThe mod $p$ cohomology of locally symmetric spaces
for definite unitary groups at infinite level is expected to realize the
mod $p$ local Langlands correspondence for $\\mathrm{GL}_n$. In particular
\, one expects the (component at $p$) of the associated Galois representat
ion to be determined by cohomology as a smooth representation. I will desc
ribe how one can establish this expectation in many cases when the local G
alois representation is Fontaine-Laffaille. This is joint work with D. Le\
, S. Morra\, C. Park and Z. Qian.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia)
DTSTART:20201005T230000Z
DTEND:20201005T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/10
DESCRIPTION:Title: The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties
\nby Will Sawin (Columbia) as part of UCLA Number Theory Seminar\n\n\n
Abstract\nFaltings proved the statement\, previously conjectured by Shafar
evich\, that there are finitely many abelian varieties of dimension $n$\,
defined over a fixed number field\, with good reduction outside a fixed fi
nite set of primes\, up to isomorphism. In joint work with Brian Lawrence\
, we prove an analogous finiteness statement for hypersurfaces in a fixed
abelian\nvariety with good reduction outside a finite set of primes. I wil
l give a broad introduction to some of the ideas in the proof\, which buil
ds on $p$-adic Hodge theory techniques from work of Lawrence and Venkatesh
as well as a little-known field of algebraic \ngeometry.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Lawrence (Chicago)
DTSTART:20201012T230000Z
DTEND:20201012T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/11
DESCRIPTION:Title: The Shafarevich conjecture for hypersurfaces in abelian varieties
\nby Brian Lawrence (Chicago) as part of UCLA Number Theory Seminar\n\
n\nAbstract\nLet $K$ be a number field\, $S$ a finite set of primes of $O_
K$\, and $g$ a positive integer. Shafarevich conjectured\, and Faltings p
roved\, that there are only finitely many curves of genus $g$\, defined ov
er $K$ and having good reduction outside $S$. Analogous results have been
proven for other families\, replacing "curves of genus $g$" with "K3 surf
aces"\, "del Pezzo surfaces" etc.\; these results are also called Shafarev
ich conjectures. There are good reasons to expect the Shafarevich conject
ure to hold for many families of varieties: the moduli space should have o
nly finitely many integral points.\n\nWill Sawin and I prove this for hype
rsurfaces in abelian varieties of dimension not equal to 3.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT/IAS)
DTSTART:20201019T230000Z
DTEND:20201019T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/12
DESCRIPTION:Title: Equivariant localization and base change functoriality\nby To
ny Feng (MIT/IAS) as part of UCLA Number Theory Seminar\n\n\nAbstract\nLa
fforgue and Genestier-Lafforgue have constructed the global and (semisim
plified) local Langlands correspondences for arbitrary reductive groups o
ver function fields. I will explain some recently established properties
of these correspondences regarding base change functoriality: existence of
transfers for mod $p$ automorphic forms through $p$-cyclic base change in
the global correspondence\, and Tate cohomology realizes $p$-cyclic base
change in the mod $p$ local correspondence. In particular\, the local stat
ement verifies a conjecture of Treumann-Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Emory (Toronto)
DTSTART:20201026T230000Z
DTEND:20201026T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/13
DESCRIPTION:Title: A multiplicity one theorem for general spin groups\nby Meliss
a Emory (Toronto) as part of UCLA Number Theory Seminar\n\n\nAbstract\nA c
lassical problem in representation theory is how a\nrepresentation of a gr
oup decomposes when restricted to a subgroup. In the\n1990s\, Gross-Prasad
formulated an intriguing conjecture regarding the\nrestriction of represe
ntations\, also known as branching laws\, of special\northogonal groups.
Gan\, Gross and Prasad extended this conjecture\, now\nknown as the local
Gan-Gross-Prasad (GGP) conjecture\, to the remaining\nclassical groups. Th
ere are many ingredients needed to prove a local GGP\nconjecture. In this
talk\, we will focus on the first ingredient: a\nmultiplicity at most one
theorem.\nAizenbud\, Gourevitch\, Rallis and Schiffmann proved a multipli
city (at\nmost) one theorem for restrictions of irreducible representation
s of\ncertain p-adic classical groups and Waldspurger proved the same theo
rem\nfor the special orthogonal groups. We will discuss work that establis
hes a\nmultiplicity (at most) one theorem for restrictions of irreducible\
nrepresentations for a non-classical group\, the general spin group. This
is\njoint work with Shuichiro Takeda.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mundy (Columbia)
DTSTART:20201110T000000Z
DTEND:20201110T005000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/14
DESCRIPTION:Title: Galois representations into $G_2$ and lattice constructions\n
by Sam Mundy (Columbia) as part of UCLA Number Theory Seminar\n\n\nAbstrac
t\nI will describe some recent work in progress on the\nsymmetric cube Blo
ch-Kato conjecture\, constructing elements in certain\nsymmetric cube Selm
er groups. This work goes by $p$-adically deforming\nEisenstein series on
the exceptional group $G_2$ in a cuspidal family\,\nand taking Galois repr
esentations associated with the members of this\nfamily. I will describe w
hat one must do on the Galois side to make\nthis method work.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amanda Folsom (Amherst College)
DTSTART:20201208T000000Z
DTEND:20201208T005000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/15
DESCRIPTION:by Amanda Folsom (Amherst College) as part of UCLA Number Theo
ry Seminar\n\n\nAbstract\nQuantum modular forms\, defined in the rationals
$\\mathbb{Q}$\, transform like modular forms do on the upper half plane $
\\mathcal{H}$\, up to suitably analytic error functions. In this talk we
give frameworks for two different examples of quantum modular forms origin
ally due to Zagier: the Dedekind sum\, and a certain $q$-hypergeometric s
um due to Kontsevich. For the first\, we extend work of Bettin and Conrey
and define twisted Eisenstein series\, study their period functions\, and
establish quantum modularity of certain cotangent-zeta sums. For the sec
ond\, we discuss results due to Hikami\, Lovejoy\, the author\, and others
\, on quantum modular and quantum Jacobi forms ultimately related to color
ed Jones polynomials for a certain family of knots.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eknath Ghate (Tata Institute)
DTSTART:20201103T030000Z
DTEND:20201103T035000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/16
DESCRIPTION:Title: Non-admissible modulo $p$ representations of $\\mathrm{GL}_2(\\ma
thbb{Q}_{p^2})$\nby Eknath Ghate (Tata Institute) as part of UCLA Numb
er Theory Seminar\n\n\nAbstract\nThe notion of admissibility of representa
tions of $p$-adic groups goes back to Harish-Chandra. Jacquet\, Bernstein
and Vigneras have shown that smooth irreducible representations of connect
ed reductive $p$-adic groups over algebraically closed fields of character
istic different from $p$ are admissible.\n\nWe use a Diamond diagram attac
hed to a $2$-dimensional reducible split mod $p$ Galois representation of
$\\mathrm{Gal}_{\\mathbb{Q}_{p^2}}$ to construct a non-admissible smooth i
rreducible mod $p$ representation of $\\mathrm{GL}_2(\\mathbb{Q}_{p^2})$ f
ollowing the approach of Daniel Le.\n\nThis is joint work with Mihir Sheth
.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Pozzi (UCL/CRM)
DTSTART:20201117T000000Z
DTEND:20201117T005000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/17
DESCRIPTION:Title: Derivatives of Hida families and rigid meromorphic cocycles\n
by Alice Pozzi (UCL/CRM) as part of UCLA Number Theory Seminar\n\n\nAbstra
ct\nA rigid meromorphic cocycle is a class in the first cohomology of the
group\n$\\mathrm{SL}_2(\\mathbb{Z}[1/p])$ acting on the non-zero rigid mer
omorphic functions on the Drinfeld\n$p$-adic upper half plane by Möbius t
ransformation. Rigid meromorphic cocycles\ncan be evaluated at points of r
eal multiplication\, and their values conjecturally\nlie in the ring class
field of real quadratic fields\, suggesting striking analogies\nwith the
classical theory of complex multiplication.\nIn this talk\, we discuss the
relation between the derivatives of certain $p$-adic\nfamilies of Hilbert
modular forms and rigid meromorphic cocycles. We explain\nhow the study o
f congruences between cuspidal and Eisenstein families allows\nus to show
the algebraicity of the values of a certain rigid meromorphic cocycle\nat
real multiplication points.\nThis is joint work with Henri Darmon and Jan
Vonk.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boya Wen (Princeton)
DTSTART:20201201T000000Z
DTEND:20201201T005000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/18
DESCRIPTION:Title: A Gross-Zagier Formula for CM cycles over Shimura Curves\nby
Boya Wen (Princeton) as part of UCLA Number Theory Seminar\n\n\nAbstract\n
In this talk I will introduce my thesis work in progress to prove a Gross-
Zagier formula for CM cycles over Shimura curves. The formula connects the
global height pairing of special cycles in Kuga varieties over Shimura cu
rves with the derivatives of the $L$-functions associated to weight-$2k$ m
odular forms. As a key original ingredient of the proof\, I will introduce
some harmonic analysis on local systems over graphs\, including an explic
it construction of Green's function\, which we apply to compute some local
intersection numbers.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josh Lam (Harvard)
DTSTART:20201124T000000Z
DTEND:20201124T005000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/19
DESCRIPTION:Title: Calabi-Yau varieties and Shimura varieties\nby Josh Lam (Harv
ard) as part of UCLA Number Theory Seminar\n\n\nAbstract\nCalabi-Yau (CY)
varieties are certain special varieties which have been the subject of int
ense studies by algebraic geometers in the last few decades. I will try to
explain some arithmetic properties of these varieties\; more specifically
\, I will discuss two results on the Attractor Conjecture which was formul
ated by Greg Moore in 1998. Throughout I will emphasize the difference bet
ween CYs with and without Shimura moduli. Time permitting\, I will discuss
what one can (conjecturally!) expect from CYs with and without Shimura mo
duli. I will not assume familiarity with CYs or Shimura varieties.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sheng-Chi Shih (Univ of Vienna)
DTSTART:20210222T190000Z
DTEND:20210222T195000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/21
DESCRIPTION:Title: Geometry of the Hilbert cuspidal eigenvariety at weight one Eisen
stein points\nby Sheng-Chi Shih (Univ of Vienna) as part of UCLA Numbe
r Theory Seminar\n\n\nAbstract\nIn this talk\, we will report on a joint w
ork with Adel Betina and Mladen\nDimitrov about the geometry of the Hilber
t cuspidal eigenvarity at a\npoint $f$ coming from a weight one Eisenstein
series irregular at a single\nprime $P$ of the totally real field $F$ abo
ve $p$.\n\nAssuming Leopoldt's conjecture for $F$ at $p$\, we show that th
e nearly\nordinary cuspidal eigenvariety is étale at f over the weight sp
ace when\n$[F_P:Q_p]\\geq[F:Q]−1$\, and hence\, the ordinary eigencurve
is étale over the\nweight space as well. When $F_P=Q_p$ we show that the
eigenvariety is\nsmooth at $f$\, while in all the remaining cases\, we pro
ve that the\neigenvariety is never smooth at $f$.\n\nIf time permits\, we
will also discuss some applications in Iwasawa\nTheory and a new proof of
the rank 1 Gross-Stark conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazim Büyükboduk (University College Dublin)
DTSTART:20210125T180000Z
DTEND:20210125T185000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/22
DESCRIPTION:Title: Perrin-Riou style critical $p$-adic $L$-functions\nby Kazim B
üyükboduk (University College Dublin) as part of UCLA Number Theory Semi
nar\n\n\nAbstract\nI will report on joint work with Denis Benois\, where w
e gave a Perrin-Riou-style construction of Bellaïche's $p$-adic $L$-funct
ion (as well as its improvements) at a $\\theta$-critical point on the eig
encurve\, with applications towards leading term formulae. Besides the int
erpolation of the Beilinson-Kato elements about this point\, the key input
to prove the interpolative properties is a new "eigenspace-transition via
differentiation" principle.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke)
DTSTART:20210119T000000Z
DTEND:20210119T005000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/23
DESCRIPTION:Title: Pointwise Bound for $\\ell$-torsion of Class Groups\nby Jiuya
Wang (Duke) as part of UCLA Number Theory Seminar\n\n\nAbstract\n$\\ell$-
torsion conjecture states that $\\ell$-torsion of the\nclass group $|\\tex
t{Cl}_K[\\ell]|$ for every number field $K$ is bounded\nby $\\text{Disc}(K
)^{\\epsilon}$. It follows from a classical result of\nBrauer-Siegel\, or
even earlier result of Minkowski that the class number\n$|\\text{Cl}_K|$ o
f a number field $K$ are always bounded by\n$\\text{Disc}(K)^{1/2+\\epsilo
n}$\, therefore we obtain a trivial bound\n$\\text{Disc}(K)^{1/2+\\epsilon
}$ on $|\\text{Cl}_K[\\ell]|$. We will talk\nabout results on this conject
ure\, and recent works on breaking the\ntrivial bound for $\\ell$-torsion
of class groups based on the work of\nEllenberg-Venkatesh.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Cass (Harvard)
DTSTART:20210302T000000Z
DTEND:20210302T005000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/24
DESCRIPTION:Title: A geometric construction of central elements in affine mod $p$ He
cke algebras\nby Robert Cass (Harvard) as part of UCLA Number Theory S
eminar\n\n\nAbstract\nLet $G$ be a split connected reductive group over a
local field of positive characteristic. In the case of characteristic zero
coefficients\, Gaitsgory gave a geometric construction of central element
s in the affine Hecke algebra of $G$ by applying a nearby cycles functor o
n a Beilinson-Drinfeld affine Grassmannian. In this talk I will explain ho
w to do an analogous construction for the affine mod $p$ Hecke algebra of
$G$. Our techniques combine the geometry of Gaitsgory's construction (and
simplifications due to Zhu) with perverse mod $p$ sheaves and tools from $
F$-singularities.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Iyengar (King's College London)
DTSTART:20210308T190000Z
DTEND:20210308T195000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/25
DESCRIPTION:Title: The Iwasawa main conjecture over the extended eigencurve\nby
Ashwin Iyengar (King's College London) as part of UCLA Number Theory Semin
ar\n\n\nAbstract\nIn this talk\, I will discuss a formulation of the Iwasa
wa main conjecture in families over the extended eigencurve\, which is an
extension of the Coleman-Mazur eigencurve into characteristic $p$. This in
volves constructing a family of $p$-adic $L$-functions\, a family of Galoi
s representations\, and showing the characteristic ideal sheaves work in f
amilies. I’ll give an overview and then give as much detail of the const
ruction as time permits.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pol van Hofton (King's College London)
DTSTART:20210329T170000Z
DTEND:20210329T175000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/26
DESCRIPTION:Title: Monodromy and irreducibility of Igusa varieties\nby Pol van H
ofton (King's College London) as part of UCLA Number Theory Seminar\n\n\nA
bstract\nIgusa varieties are smooth varieties in characteristic $p$ arisin
g naturally as covers of certain subvarieties (central leaves) of Shimura
varieties\, for example of the ordinary locus of the modular curve. The $\
\ell$-adic cohomology of Igusa varieties acts as a bridge between the coho
mology of Rapoport-Zink spaces (local) and the cohomology of Shimura varie
ties (global)\, and it is therefore very interesting to study this cohomol
ogy. In this talk I will discuss recent joint work with Luciena Xiao Xiao\
, where we compute the 0th cohomology group. This is equivalent to determi
ning the irreducible components of Igusa varieties\, and our results gener
alise results of Hida and Chai-Oort. Our strategy combines recent work of
D’Addezio on monodromy of compatible local systems with a generalisation
of a method of Hida\, and the Honda-Tate theory for Shimura varieties of
Hodge type of Kisin--Madapusi Pera--Shin.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sally Gilles (Imperial College London)
DTSTART:20210412T170000Z
DTEND:20210412T175000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/27
DESCRIPTION:Title: Syntomic cohomology and period morphisms\nby Sally Gilles (Im
perial College London) as part of UCLA Number Theory Seminar\n\n\nAbstract
\nIn 2017\, Colmez and Niziol proved a comparison theorem between arithmet
ic $p$-adic nearby cycles and syntomic cohomology sheaves. To prove it\, t
hey gave a local construction using $(\\phi\, \\Gamma)$-modules theory whi
ch allows to reduce the period isomorphism to a comparison theorem between
cohomologies of Lie algebras. I will explain the geometric version of thi
s local construction and how to globalize it to get a new period isomorphi
sm. In particular\, the explicit description of this new isomorphism can b
e used to compare previous constructions of period morphisms and prove the
y are equal.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhixiang Wu (Université Paris-Saclay)
DTSTART:20210419T230000Z
DTEND:20210419T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/28
DESCRIPTION:Title: Companion forms and partially classical eigenvarieties\nby Zh
ixiang Wu (Université Paris-Saclay) as part of UCLA Number Theory Seminar
\n\n\nAbstract\nIn general\, there exist $p$-adic automorphic forms of dif
ferent weights with the same associated $p$-adic Galois representation. In
this talk\, I will report some result on the existence of companion forms
for definite unitary groups when the Hodge-Tate weights are not regular\,
generalizing the work of Breuil-Hellmann-Schraen in regular cases. One ke
y ingredient of the proof is some partially classical properties of $p$-ad
ic automorphic forms in the term of Emerton's Jacquet module for locally a
nalytic representations\, which will imply some partially de Rham properti
es of Galois representations in finite slope cases with the help of Ding's
partial eigenvarieties.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART:20210426T230000Z
DTEND:20210426T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/29
DESCRIPTION:Title: Kolyvagin's conjecture and higher congruences of modular forms\nby Naomi Sweeting (Harvard) as part of UCLA Number Theory Seminar\n\n\n
Abstract\nGiven an elliptic curve $E$\, Kolyvagin used CM points on modul
ar curves to construct a system of classes valued in the Galois cohomology
of the torsion points of $E$. Under the conjecture that not all of these
classes vanish\, he gave a description for the Selmer group of $E$. This
talk will report on recent work proving new cases of Kolyvagin's conjectur
e. The proof builds on work of Wei Zhang\, who used congruences between mo
dular forms to prove Kolyvagin's conjecture under some technical hypothese
s. We remove many of these hypotheses by considering congruences modulo hi
gher powers of $p$. The talk will explain the difficulties associated with
higher congruences of modular forms and how they can be overcome\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guido Kings (Universität Regensburg)
DTSTART:20210510T170000Z
DTEND:20210510T175000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/30
DESCRIPTION:Title: Equivariant Eisenstein classes\, critical values of Hecke $L$-fun
ctions and $p$-adic interpolation\nby Guido Kings (Universität Regens
burg) as part of UCLA Number Theory Seminar\n\n\nAbstract\nI report on joi
nt work with Johannes Sprang. Let $K$ be a CM field and\n$L/K$ be an exten
sion of degree $n$ and $\\chi$ be an algebraic critical Hecke\ncharacter o
f $L$. Then we show that the $L$-value $L(\\chi\, 0)$ divided by\ncarefull
y normalized Shimura-Katz periods is integral and construct a\n$p$-adic $L
$-function for $\\chi$. This generalizes results by Damerell\, Shimura and
Katz for CM fields ($L = K$) and settles all open cases of algebraicity f
or critical Hecke $L$-values.\n\nOur method relies on a detailed analysis
of new equivariant motivic Eisenstein classes and especially on the study
of their de Rham realizations and is completely different from the classic
al approach by Shimura and Katz. The de Rham realization of these Eisenste
in classes\ncan be explicitly described in terms of Eisenstein-Kronecker s
eries and the equivariant setting is crucial to connect them with the $L$-
function of $\\chi$. An integral refinement of this construction leads dir
ectly to a geometric construction of a $p$-adic measure without any need t
o check congruences for the Eisenstein series.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaoyu Zhang (Universität Duisburg-Essen)
DTSTART:20210503T170000Z
DTEND:20210503T175000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/31
DESCRIPTION:Title: CM points on compact orthogonal groups and theta lifts mod $p$\nby Xiaoyu Zhang (Universität Duisburg-Essen) as part of UCLA Number Th
eory Seminar\n\n\nAbstract\nCM points on Shimura varieties are very useful
in the study of arithmetic properties of automorphic forms\, $L$-function
s\, etc.. C. Cornut and V. Vatsal proved certain dynamical properties of t
hese points on quaternions and then deduce Mazur's conjectures as well as
study the non-vanishing of Rankin-Selberg $L$-values. In this talk we try
to follow the strategy of Cornut and Vatsal and generalise their result to
certain compact orthogonal groups and as an application\, we study the no
n-vanishing problem of theta lifts mod $p$ from orthogonal group to symple
ctic group.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Hough (Stony Brook)
DTSTART:20211004T220000Z
DTEND:20211004T225000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/32
DESCRIPTION:Title: Recent developments in orbit counting methods\nby Robert Houg
h (Stony Brook) as part of UCLA Number Theory Seminar\n\n\nAbstract\nBharg
ava pioneered methods from the geometry of numbers to count integral orbit
s in representation spaces ordered by invariants. I will discuss recent a
nalytic techniques in development to strengthen the methods\, including sp
ectral expansion of the underlying homogeneous spaces\, classification of
local orbits and their finite Fourier transforms\, and subconvex estimates
for the enumerating zeta functions. In particular\, we have obtained a s
trong answer to a question of Bhargava and Gross at a conference at the Am
erican Institute of Math explaining a barrier to equidistribution in the s
hape of cubic fields by obtaining poles and residues in the zeta function
enumerating the Weyl sums in the Eisenstein spectrum. Joint work with Eun
Hye Lee.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Florea (UC Irvine)
DTSTART:20211018T220000Z
DTEND:20211018T225000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/33
DESCRIPTION:Title: The Ratios Conjecture over function fields\nby Alexandra Flor
ea (UC Irvine) as part of UCLA Number Theory Seminar\n\n\nAbstract\nI will
talk about some recent joint work with H. Bui and J. Keating where we stu
dy the Ratios Conjecture for the family of quadratic L-functions over func
tion fields. I will also discuss the closely related problem of obtaining
upper bounds for negative moments of L-functions\, which allows us to obta
in partial results towards the Ratios Conjecture in the case of one over o
ne\, two over two and three over three L-functions.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Horawa (Michigan)
DTSTART:20211011T220000Z
DTEND:20211011T225000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/34
DESCRIPTION:Title: Motivic action on coherent cohomology of Hilbert modular varietie
s\nby Alex Horawa (Michigan) as part of UCLA Number Theory Seminar\n\n
\nAbstract\nA surprising property of cohomology of locally symmetric space
s is that Hecke operators can act on multiple cohomological degrees with t
he same eigenvalues. We will discuss this phenomenon for coherent cohomolo
gy of line bundles on modular curves and\, more generally\, Hilbert modula
r varieties. We propose an arithmetic explanation: a hidden degree-shiftin
g action of a certain motivic cohomology group (the Stark unit group). Thi
s extends the conjectures of Venkatesh\, Prasanna\, and Harris to Hilbert
modular varieties.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashay Burungale (Caltech)
DTSTART:20211025T220000Z
DTEND:20211025T225000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/35
DESCRIPTION:Title: p-converse to a theorem of Gross-Zagier and Kolyvagin - dihedral
primes\nby Ashay Burungale (Caltech) as part of UCLA Number Theory Sem
inar\n\nLecture held in Math Science Building 5203.\n\nAbstract\nSuch a p-
converse will be outlined (joint with Chris Skinner).\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anwesh Ray (Univ of British Columbia)
DTSTART:20211101T220000Z
DTEND:20211101T225000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/36
DESCRIPTION:Title: Iwasawa theory and congruences for the symmetric square of a modu
lar form\nby Anwesh Ray (Univ of British Columbia) as part of UCLA Num
ber Theory Seminar\n\n\nAbstract\nI will report on joint work with R. Suja
tha and V. Vatsal. Two\n$p$-ordinary Hecke-eigenforms are are congruent at
a prime $\\varpi|p$ if\nall but finitely many of their Fourier coefficien
ts are congruent modulo\n$\\varpi$. R. Greenberg and V. Vatsal showed in 2
000 that the\nIwasawa-invariants of congruent modular forms are related. A
s a result\, if\n$\\mu$-invariant vanishes and the main conjecture holds f
or a given\nHecke-eigenform\, then the same is true for a congruent Hecke-
eigenform.\nThis involves studying the behavior of Selmer groups and p-adi
c L-functions\nwith respect to congruences. We generalize these results to
symmetric\nsquare representations.\n\n The main task at hand is that the
p-adic L-functions for the symmetric\nsquare exhibit congruences. In this
setting\, the normalized L-values for\n$sym^2(f)$ can be expressed in term
s of the Petersson inner product of $f$\nwith a nearly holomorphic functio
n. This function is expressed as the\nproduct of a theta function and an E
isentein series. The ordinary\nholomorphic projection of this function is
shown to have nice properties.\nThe Petersson inner product is modified an
d related to an abstractly\ndefined algebraic pairing due to Hida\, and th
e two pairing are related up\nto a "canonical period". Under further hypot
heses\, it is shown that this\ncanonical period is suitably well behaved.
For this\, we assume a certain\nversion of Ihara's lemma\, which is known
in certain cases.\n\n With these preparations\, we are able to show that n
ormalized L-values for\nthe symmetric square behave well with respect to c
ongruence\, and hence\, the\np-adic L-functions too. It follows that the a
nalytic Iwasawa invariants for congruent Hecke-eigencuspforms are related.
Such results for the algebraic Iwasawa invariants follow from work of R.
Greenberg and V. Vatsal. Just as in the classicial case\, the results have
implications to the main conjecture. If time permits\, we will introduce
the role of the fine-Selmer\ngroup and discuss a condition for the vanishi
ng on the $\\mu$-invariant that\ncan be stated purely in terms of the resi
dual representation.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brandon Alberts (UC San Diego)
DTSTART:20211129T230000Z
DTEND:20211129T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/37
DESCRIPTION:Title: Refining Malle's Conjecture for Inductive Counting Methods\nb
y Brandon Alberts (UC San Diego) as part of UCLA Number Theory Seminar\n\n
Lecture held in Math Science Building 5203.\n\nAbstract\nMalle's conjectur
e predicts the asymptotic growth rate of the number of G-extensions F/K of
a number field K with absolute discriminant bounded above by X\, where X
tends towards infinity. I will discuss a joint project with Robert Lemke O
liver\, Jiuya Wang\, and Melanie Matchett Wood to approach this conjecture
inductively by first restricting to G-extensions F/K containing a fixed i
ntermediate extension L/K\, then taking a sum over choices of intermediate
extensions. A fundamental concept in this talk will be the related questi
on of finding the distribution of elements of the first Galois cohomology
group\, $H^1(K\,T)$. In particular\, I will address a joint paper with Eva
n O'Dorney using harmonic analysis to study $H^1(K\,T)$.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yong Suk Moon (Univ of Arizona)
DTSTART:20211108T230000Z
DTEND:20211108T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/38
DESCRIPTION:Title: Prismatic crystals and crystalline representations in the relativ
e case\nby Yong Suk Moon (Univ of Arizona) as part of UCLA Number Theo
ry Seminar\n\n\nAbstract\nLet k be a perfect field of characteristic p > 2
\, and let K be a finite totally ramified extension of W(k)[1/p]. Bhatt-Sc
holze recently proved that the category of prismatic F-crystals on the abs
olute prismatic site over O_K is equivalent to the category of lattices of
crystalline representations of G_K. We study an analogous situation in th
e relative case. Let Spf R be an affine p-adic formal scheme smooth over O
_K. We show there is a natural faithful functor from the category of certa
in completed F-crystals on the absolute prismatic site over R to the categ
ory of crystalline Z_p-local systems on the generic fiber of Spf R. Furthe
rmore\, we show the functor gives an equivalence when R is a formal torus
over O_K. This is a joint work with Heng Du\, Tong Liu\, Koji Shimizu.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gilyoung Cheong (UC Irvine)
DTSTART:20211115T230000Z
DTEND:20211115T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/39
DESCRIPTION:Title: Counting square-free numbers in arithmetic geometry\nby Gilyo
ung Cheong (UC Irvine) as part of UCLA Number Theory Seminar\n\n\nAbstract
\nWe delve into an innocuous question about counting\n"square-free numbers
" in various forms\, following the philosophy of\nWeil's three columns.\n\
nExample 1. We count square-free integers.\n\nExample 2. We count square-f
ree polynomials of a fixed degree over a\nfinite field.\n\nExample 3. We c
ompute the Betti numbers of the space of square-free\npolynomials of a fix
ed degree over complex numbers by quoting a theorem\nof Arnol'd in topolog
y.\n\nBy viewing Example 3 as counting square-free 0-cycles on the affine
line\nover complex numbers\, we add one more example to this list\, using
our\nmain result.\n\nExample 4. We compute the Betti numbers of the space
of square-free\n0-cycles of a fixed degree on a punctured elliptic curve o
ver complex\nnumbers.\n\nWe briefly explain how Examples 3 and 4 can be ob
tained by showing that\nthe mixed Hodge structure of the i-th singular coh
omology group with\nrational coefficients is pure of some weight different
from i. This is\njoint work with Yifeng Huang.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UC San Diego)
DTSTART:20211122T230000Z
DTEND:20211122T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/40
DESCRIPTION:Title: A Cohen-Zagier modular form on G_2\nby Aaron Pollack (UC San
Diego) as part of UCLA Number Theory Seminar\n\nLecture held in Math Scien
ce Building 5203.\n\nAbstract\nI will report on joint work-in-progress wit
h Spencer Leslie where we define an analogue of the Cohen-Zagier Eisenstei
n series to the exceptional group G_2. Recall that the Cohen-Zagier Eisens
tein series is a weight 3/2 modular form whose Fourier coefficients see th
e class numbers of imaginary quadratic fields. We define a particular modu
lar form of weight 1/2 on G_2\, and prove that its Fourier coefficients se
e the 2-torsion in the narrow class groups of totally real cubic fields. I
n particular: 1) we define a notion of modular forms of half-integral weig
ht on certain exceptional groups\, 2) we prove that these modular forms ha
ve a nice theory of Fourier coefficients\, and 3) we partially compute the
Fourier coefficients of a particular nice example on G_2.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Lawrence (UCLA)
DTSTART:20220110T230000Z
DTEND:20220110T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/41
DESCRIPTION:Title: Sparsity of Integral Points on Moduli Spaces of Varieties\nby
Brian Lawrence (UCLA) as part of UCLA Number Theory Seminar\n\n\nAbstract
\nInteresting moduli spaces don't have many integral points. More precisel
y\, if X is a variety over a number field\, admitting a variation of Hodge
structure whose associate period map is injective\, then the number of S-
integral points on X of height at most H grows more slowly than H^{\\epsil
on}\, for any positive \\epsilon. This is a sort of weak generalization of
the Shafarevich conjecture\; it is a consequence of a point-counting theo
rem of Broberg\, and the largeness of the fundamental group of X. Joint wi
th Ellenberg and Venkatesh.\n\nhttps://arxiv.org/abs/2109.01043\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Harris (Columbia)
DTSTART:20220112T230000Z
DTEND:20220112T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/42
DESCRIPTION:Title: L-functions and periods of automorphic forms\nby Michael Harr
is (Columbia) as part of UCLA Number Theory Seminar\n\n\nAbstract\nThis is
a report on recent work with Grobner and Lin on the critical values of\nR
ankin-Selberg L-functions of GL(n)xGL(n-1) over CM fields. By reinterpre
ting these critical values in terms of automorphic periods of holomorphic
automorphic forms on unitary groups\, we show that the automorphic periods
of holomorphic forms can be factored as products of coherent cohomologica
l forms\, compatibly with a motivic factorization predicted by the Tate co
njecture. All of these results are conditional on a conjecture on non-vani
shing of twists of automorphic L-functions of GL(n) by anticyclotomic char
acters of finite order\, and are stated under a certain regularity conditi
on.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hang Xue (Univ. of Arizona)
DTSTART:20220124T230000Z
DTEND:20220124T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/43
DESCRIPTION:Title: The local Gan--Gross--Prasad conjecture for real unitary groups\nby Hang Xue (Univ. of Arizona) as part of UCLA Number Theory Seminar\n
\n\nAbstract\nI explain a very simple proof of the local Gan--Gross--Prasa
d conjecture for real unitary groups. I also explain meromorphic continuat
ion of the explicit tempered intertwining map based on a similar idea.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UC Santa Barbara)
DTSTART:20220131T230000Z
DTEND:20220131T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/44
DESCRIPTION:Title: Selmer classes on CM elliptic curves of rank 2\nby Francesc C
astella (UC Santa Barbara) as part of UCLA Number Theory Seminar\n\nLectur
e held in Math Science 5118.\n\nAbstract\nLet E be an elliptic curve over
Q\, and let p be a prime of good ordinary reduction for E. Following the p
ioneering work of Skinner (and independently Wei Zhang) from about 8 years
ago\, there is a growing number of results in the direction of a p-conver
se to a theorem of Gross-Zagier and Kolyvagin\, showing that if the p-adic
Selmer group of E is 1-dimensional\, then a Heegner point on E has infini
te order. In this talk\, I'll report on the proof of an analogue of Skinne
r's result in the rank 2 case\, in which Heegner points are replaced by ce
rtain generalized Kato classes introduced by Darmon-Rotger. For E without
CM\, such an analogue was obtained in an earlier work with M.-L. Hsieh\, a
nd in this talk I'll focus on the CM case\, whose proof uses a different s
et of ideas.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Dunn (Caltech)
DTSTART:20220207T230000Z
DTEND:20220207T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/45
DESCRIPTION:Title: Bias in cubic Gauss sums: Patterson's conjecture\nby Alex Dun
n (Caltech) as part of UCLA Number Theory Seminar\n\nLecture held in Math
Science 5118.\n\nAbstract\nWe prove\, in this joint work with Maksym Radzi
will\, a 1978 conjecture of S. Patterson (conditional on the Generalised R
iemann hypothesis) concerning the bias of cubic Gauss sums. This explains
a well-known numerical bias in the distribution of cubic Gauss sums first
observed by Kummer in 1846.\n\nThere are two important byproducts of our p
roof. The first is an explicit level aspect Voronoi summation formula for
cubic Gauss sums\, extending computations of Patterson and Yoshimoto. Seco
ndly\, we show that Heath-Brown's cubic large sieve is sharp under GRH. Th
is disproves the popular belief that the cubic large sieve can be improved
.\n\nAn important ingredient in our proof is a dispersion estimate for cub
ic Gauss sums. It can be interpreted as a cubic large sieve with correctio
n by a non-trivial asymptotic main term. This estimate relies on the Gener
alised Riemann Hypothesis\, and is one of the fundamental reasons why our
result is conditional.\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mingjie Chen (UC San Diego)
DTSTART:20220214T230000Z
DTEND:20220214T235000Z
DTSTAMP:20260404T094753Z
UID:UCLA_NTS/46
DESCRIPTION:Title: Orienteering with one endomorphism\nby Mingjie Chen (UC San D
iego) as part of UCLA Number Theory Seminar\n\n\nAbstract\nSupersingular i
sogeny-based cryptosystems are strong contenders for post-quantum cryptogr
aphy standardization. Such cryptosystems rely on the hardness of path-find
ing on supersingular isogeny graphs. The path-finding problem is known to
reduce to the endomorphism ring problem. Can path-finding be reduced to kn
owing just one endomorphism? In this talk\, we give explicit classical and
quantum algorithms for path-finding to an initial curve using the knowled
ge of one endomorphism. An endomorphism gives an orientation of a supersin
gular elliptic curve. We use the theory of oriented supersingular isogeny
graphs and algorithms for taking ascending/descending/horizontal steps on
such graphs.\n\npaper link: https://arxiv.org/pdf/2201.11079.pdf\n
LOCATION:https://stable.researchseminars.org/talk/UCLA_NTS/46/
END:VEVENT
END:VCALENDAR