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BEGIN:VEVENT
SUMMARY:Romyar Sharifi (UCLA)
DTSTART:20200413T230000Z
DTEND:20200413T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/1
DESCRIPTION:Title: Eisenstein cocycles in motivic cohomology\nby Romyar Sharifi (UCLA) a
s 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 multiplica
tive group over the rationals and a universal elliptic curve. Focusing on
the first case\, I’ll explain how the cocycle we construct specializes
to homomorphisms taking modular symbols for congruence subgroups to specia
l elements in second cohomology groups of cyclotomic fields and satisfies
a certain property of being “Eisenstein”.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jayce Getz (Duke)
DTSTART:20200427T230000Z
DTEND:20200427T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/2
DESCRIPTION:Title: On triple product L-functions\nby Jayce Getz (Duke) as part of UCLA N
umber Theory Seminar\n\n\nAbstract\nEstablishing the conjectured analytic
properties of triple product L-functions is a crucial case of Langlands fu
nctoriality. However\, little is known. I will present work in progress on
the case of triples of automorphic representations on GL_3\; in some sens
e this is the smallest case that appears out of reach via standard techniq
ues. The approach involves a relative trace formula and Poisson summation
on spherical varieties in the sense of Braverman-Kazhdan\, Ngo\, and Sakel
laridis.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zheng Liu (UC Santa Barbara)
DTSTART:20200518T230000Z
DTEND:20200518T235000Z
DTSTAMP:20260422T225841Z
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\nAbstract
\nIn order to verify the compatibility between the conjecture of Coates--P
errin-Riou and the interpolation results of the $p$-adic $L$-functions con
structed by using the doubling method\, a doubling archimedean zeta integr
al needs to be calculated for holomorphic discrete series. When the holomo
rphic discrete series is of scalar weight\, it has been done by Bocherer-S
chmidt and Shimura. In this talk\, I will explain a way to compute this ar
chimedean zeta integral for unitary groups of arbitrary signatures and gen
eral vector weights. This is a joint work with Ellen Eischen.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zijian Yao (Harvard)
DTSTART:20200527T230000Z
DTEND:20200527T235000Z
DTSTAMP:20260422T225841Z
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\nAbstract\n
A major goal of $p$-adic Hodge theory is to relate arithmetic structures c
oming from various cohomology theories of $p$-adic varieties. Such compari
sons are usually achieved by constructing intermediate cohomology theories
. A somewhat recent successful theory\, namely the $\\mathrm{A}_\\mathrm{i
nf}$-cohomology\, has been invented by Bhatt--Morrow--Scholze\, originally
via perfectoid spaces. In this talk\, I will describe a simpler approach
to prove the comparison between $\\mathrm{A}_\\mathrm{inf}$-cohomology and
the crystalline cohomology over Fontaine's period ring $\\mathrm{A}_\\mat
hrm{cris}$\, using flat descent of cotangent complexes. This approach also
allows us to prove compatibilities of certain $p$-adic filtrations. Time
permitting\, I will discuss some work in progress (partially joint with Ha
nsheng Diao) in the semistable/logarithmic case.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Stubley (UChicago)
DTSTART:20200601T230000Z
DTEND:20200601T235000Z
DTSTAMP:20260422T225841Z
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 decompositio
n group at $p$. A natural question to ask is under what conditions this up
per triangular decomposition splits as a direct sum. Ghate and Vatsal 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 their proo
f\, the weight one members of the family play a key role.\n\nI'll talk abo
ut work in progress which aims to answer similar questions in the case of
Galois representations for a totally real field which are split at only so
me of the decomposition groups at primes above $p$. In this work Hilbert m
odular forms of partial weight one play a central role\; I'll discuss what
is known about them and to what extent the techniques of Ghate and Vatsal
can be adapted to this situation.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (University of Cambridge)
DTSTART:20200420T220000Z
DTEND:20200420T225000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/6
DESCRIPTION:Title: Symmetric power functoriality for holomorphic modular forms\nby Jack
Thorne (University of Cambridge) as part of UCLA Number Theory Seminar\n\n
\nAbstract\nLanglands’s functoriality conjectures predict the existence
of "liftings" of automorphic representations along morphisms of L-groups.
A basic case of interest comes from the irreducible algebraic representati
ons 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 existence
of the corresponding functorial liftings for a broad class of holomorphic
modular forms\, including Ramanujan’s Delta function.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bas Edixhoven (Universiteit Leiden)
DTSTART:20200504T180000Z
DTEND:20200504T185000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/7
DESCRIPTION:Title: Geometric quadratic Chabauty\nby Bas Edixhoven (Universiteit Leiden)
as part of UCLA Number Theory Seminar\n\n\nAbstract\nJoint work with Guido
Lido (see arxiv preprint). Determining all rational points on a curve of
genus at least 2 can be difficult. Chabauty's method (1941) is to intersec
t\, 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 points o
f the curve. If the Mordell-Weil rank is less than the genus then this met
hod has never failed. Minhyong Kim's non-abelian Chabauty programme aims t
o remove the condition on the rank. The simplest case\, called quadratic C
habauty\, was developed by Balakrishnan\, Dogra\, Mueller\, Tuitman and Vo
nk\, and applied in a tour de force to the so-called cursed curve (rank an
d genus both 3). Our work aims to make the quadratic Chabauty method small
and geometric again\, by describing it in terms of only "simple algebraic
geometry" (line bundles over the jacobian and models over the integers).\
n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michele Fornea (Princeton)
DTSTART:20200511T230000Z
DTEND:20200511T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/8
DESCRIPTION:Title: Hirzebruch-Zagier classes and rational elliptic curves over quintic field
s\nby Michele Fornea (Princeton) as part of UCLA Number Theory Seminar
\n\n\nAbstract\nIn joint work with Zhaorong Jin\, we establish new instanc
es of the BSD-conjecture for rational elliptic curves over quintic fields.
\nThe proof is p-adic in nature and relies on the comparison of two rigid
analytic functions: the automorphic p-adic L-function retaining informati
on about special L-values\, and the motivic p-adic L-function arising from
cycles on Shimura threefolds. The construction of the latter function is
of independent interest as it could be applied to other settings related t
o the Gan-Gross-Prasad conjecture.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bao Le Hung (Northwestern)
DTSTART:20200928T230000Z
DTEND:20200928T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/9
DESCRIPTION:Title: Moduli of Fontaine-Laffaille modules and mod $p$ local-global compatibili
ty\nby Bao Le Hung (Northwestern) as part of UCLA Number Theory Semina
r\n\n\nAbstract\nThe mod $p$ cohomology of locally symmetric spaces for de
finite 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 representation to
be determined by cohomology as a smooth representation. I will describe ho
w one can establish this expectation in many cases when the local Galois r
epresentation is Fontaine-Laffaille. This is joint work with D. Le\, S. Mo
rra\, C. Park and Z. Qian.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia)
DTSTART:20201005T230000Z
DTEND:20201005T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/10
DESCRIPTION:Title: The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties\nb
y Will Sawin (Columbia) as part of UCLA Number Theory Seminar\n\n\nAbstrac
t\nFaltings proved the statement\, previously conjectured by Shafarevich\,
that there are finitely many abelian varieties of dimension $n$\, defined
over a fixed number field\, with good reduction outside a fixed finite se
t of primes\, up to isomorphism. In joint work with Brian Lawrence\, we pr
ove an analogous finiteness statement for hypersurfaces in a fixed abelian
\nvariety with good reduction outside a finite set of primes. I will give
a broad introduction to some of the ideas in the proof\, which builds on $
p$-adic Hodge theory techniques from work of Lawrence and Venkatesh as wel
l as a little-known field of algebraic \ngeometry.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Lawrence (Chicago)
DTSTART:20201012T230000Z
DTEND:20201012T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/11
DESCRIPTION:Title: The Shafarevich conjecture for hypersurfaces in abelian varieties\nb
y Brian Lawrence (Chicago) as part of UCLA Number Theory Seminar\n\n\nAbst
ract\nLet $K$ be a number field\, $S$ a finite set of primes of $O_K$\, an
d $g$ a positive integer. Shafarevich conjectured\, and Faltings proved\,
that there are only finitely many curves of genus $g$\, defined over $K$
and having good reduction outside $S$. Analogous results have been proven
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://researchseminars.org/talk/UCLA_NTS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Feng (MIT/IAS)
DTSTART:20201019T230000Z
DTEND:20201019T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/12
DESCRIPTION:Title: Equivariant localization and base change functoriality\nby Tony Feng
(MIT/IAS) as part of UCLA Number Theory Seminar\n\n\nAbstract\nLafforgue
and Genestier-Lafforgue have constructed the global and (semisimplified
) local Langlands correspondences for arbitrary reductive groups over func
tion fields. I will explain some recently established properties of these
correspondences regarding base change functoriality: existence of transfe
rs for mod $p$ automorphic forms through $p$-cyclic base change in the glo
bal correspondence\, and Tate cohomology realizes $p$-cyclic base change i
n the mod $p$ local correspondence. In particular\, the local statement ve
rifies a conjecture of Treumann-Venkatesh.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Emory (Toronto)
DTSTART:20201026T230000Z
DTEND:20201026T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/13
DESCRIPTION:Title: A multiplicity one theorem for general spin groups\nby Melissa Emory
(Toronto) as part of UCLA Number Theory Seminar\n\n\nAbstract\nA classica
l problem in representation theory is how a\nrepresentation of a group dec
omposes when restricted to a subgroup. In the\n1990s\, Gross-Prasad formul
ated an intriguing conjecture regarding the\nrestriction of representation
s\, also known as branching laws\, of special\northogonal groups. Gan\, G
ross and Prasad extended this conjecture\, now\nknown as the local Gan-Gro
ss-Prasad (GGP) conjecture\, to the remaining\nclassical groups. There 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 theore
m.\nAizenbud\, Gourevitch\, Rallis and Schiffmann proved a multiplicity (a
t\nmost) one theorem for restrictions of irreducible representations of\nc
ertain p-adic classical groups and Waldspurger proved the same theorem\nfo
r the special orthogonal groups. We will discuss work that establishes a\n
multiplicity (at most) one theorem for restrictions of irreducible\nrepres
entations for a non-classical group\, the general spin group. This is\njoi
nt work with Shuichiro Takeda.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Mundy (Columbia)
DTSTART:20201110T000000Z
DTEND:20201110T005000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/14
DESCRIPTION:Title: Galois representations into $G_2$ and lattice constructions\nby Sam
Mundy (Columbia) as part of UCLA Number Theory Seminar\n\n\nAbstract\nI wi
ll describe some recent work in progress on the\nsymmetric cube Bloch-Kato
conjecture\, constructing elements in certain\nsymmetric cube Selmer grou
ps. This work goes by $p$-adically deforming\nEisenstein series on the exc
eptional group $G_2$ in a cuspidal family\,\nand taking Galois representat
ions associated with the members of this\nfamily. I will describe what one
must do on the Galois side to make\nthis method work.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amanda Folsom (Amherst College)
DTSTART:20201208T000000Z
DTEND:20201208T005000Z
DTSTAMP:20260422T225841Z
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://researchseminars.org/talk/UCLA_NTS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eknath Ghate (Tata Institute)
DTSTART:20201103T030000Z
DTEND:20201103T035000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/16
DESCRIPTION:Title: Non-admissible modulo $p$ representations of $\\mathrm{GL}_2(\\mathbb{Q}
_{p^2})$\nby Eknath Ghate (Tata Institute) as part of UCLA Number Theo
ry Seminar\n\n\nAbstract\nThe notion of admissibility of representations o
f $p$-adic groups goes back to Harish-Chandra. Jacquet\, Bernstein and Vig
neras have shown that smooth irreducible representations of connected redu
ctive $p$-adic groups over algebraically closed fields of characteristic d
ifferent from $p$ are admissible.\n\nWe use a Diamond diagram attached to
a $2$-dimensional reducible split mod $p$ Galois representation of $\\math
rm{Gal}_{\\mathbb{Q}_{p^2}}$ to construct a non-admissible smooth irreduci
ble mod $p$ representation of $\\mathrm{GL}_2(\\mathbb{Q}_{p^2})$ followin
g the approach of Daniel Le.\n\nThis is joint work with Mihir Sheth.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alice Pozzi (UCL/CRM)
DTSTART:20201117T000000Z
DTEND:20201117T005000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/17
DESCRIPTION:Title: Derivatives of Hida families and rigid meromorphic cocycles\nby Alic
e Pozzi (UCL/CRM) as part of UCLA Number Theory Seminar\n\n\nAbstract\nA r
igid 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 meromorphi
c functions on the Drinfeld\n$p$-adic upper half plane by Möbius transfor
mation. Rigid meromorphic cocycles\ncan be evaluated at points of real mul
tiplication\, and their values conjecturally\nlie in the ring class field
of real quadratic fields\, suggesting striking analogies\nwith the classic
al theory of complex multiplication.\nIn this talk\, we discuss the relati
on between the derivatives of certain $p$-adic\nfamilies of Hilbert modula
r forms and rigid meromorphic cocycles. We explain\nhow the study of congr
uences between cuspidal and Eisenstein families allows\nus to show the alg
ebraicity of the values of a certain rigid meromorphic cocycle\nat real mu
ltiplication points.\nThis is joint work with Henri Darmon and Jan Vonk.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boya Wen (Princeton)
DTSTART:20201201T000000Z
DTEND:20201201T005000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/18
DESCRIPTION:Title: A Gross-Zagier Formula for CM cycles over Shimura Curves\nby Boya We
n (Princeton) as part of UCLA Number Theory Seminar\n\n\nAbstract\nIn 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 curves wi
th the derivatives of the $L$-functions associated to weight-$2k$ modular
forms. As a key original ingredient of the proof\, I will introduce some h
armonic analysis on local systems over graphs\, including an explicit cons
truction of Green's function\, which we apply to compute some local inters
ection numbers.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josh Lam (Harvard)
DTSTART:20201124T000000Z
DTEND:20201124T005000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/19
DESCRIPTION:Title: Calabi-Yau varieties and Shimura varieties\nby Josh Lam (Harvard) as
part of UCLA Number Theory Seminar\n\n\nAbstract\nCalabi-Yau (CY) varieti
es are certain special varieties which have been the subject of intense st
udies by algebraic geometers in the last few decades. I will try to explai
n some arithmetic properties of these varieties\; more specifically\, I wi
ll discuss two results on the Attractor Conjecture which was formulated by
Greg Moore in 1998. Throughout I will emphasize the difference between CY
s with and without Shimura moduli. Time permitting\, I will discuss what o
ne can (conjecturally!) expect from CYs with and without Shimura moduli. I
will not assume familiarity with CYs or Shimura varieties.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sheng-Chi Shih (Univ of Vienna)
DTSTART:20210222T190000Z
DTEND:20210222T195000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/21
DESCRIPTION:Title: Geometry of the Hilbert cuspidal eigenvariety at weight one Eisenstein p
oints\nby Sheng-Chi Shih (Univ of Vienna) as part of UCLA Number Theor
y Seminar\n\n\nAbstract\nIn this talk\, we will report on a joint work wit
h Adel Betina and Mladen\nDimitrov about the geometry of the Hilbert cuspi
dal eigenvarity at a\npoint $f$ coming from a weight one Eisenstein series
irregular at a single\nprime $P$ of the totally real field $F$ above $p$.
\n\nAssuming Leopoldt's conjecture for $F$ at $p$\, we show that the nearl
y\nordinary cuspidal eigenvariety is étale at f over the weight space whe
n\n$[F_P:Q_p]\\geq[F:Q]−1$\, and hence\, the ordinary eigencurve is éta
le over the\nweight space as well. When $F_P=Q_p$ we show that the eigenva
riety is\nsmooth at $f$\, while in all the remaining cases\, we prove that
the\neigenvariety is never smooth at $f$.\n\nIf time permits\, we will al
so discuss some applications in Iwasawa\nTheory and a new proof of the ran
k 1 Gross-Stark conjecture.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazim Büyükboduk (University College Dublin)
DTSTART:20210125T180000Z
DTEND:20210125T185000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/22
DESCRIPTION:Title: Perrin-Riou style critical $p$-adic $L$-functions\nby Kazim Büyükb
oduk (University College Dublin) as part of UCLA Number Theory Seminar\n\n
\nAbstract\nI will report on joint work with Denis Benois\, where we gave
a Perrin-Riou-style construction of Bellaïche's $p$-adic $L$-function (as
well as its improvements) at a $\\theta$-critical point on the eigencurve
\, with applications towards leading term formulae. Besides the interpolat
ion of the Beilinson-Kato elements about this point\, the key input to pro
ve the interpolative properties is a new "eigenspace-transition via differ
entiation" principle.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuya Wang (Duke)
DTSTART:20210119T000000Z
DTEND:20210119T005000Z
DTSTAMP:20260422T225841Z
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 $|\\text{Cl}_K
[\\ell]|$ for every number field $K$ is bounded\nby $\\text{Disc}(K)^{\\ep
silon}$. It follows from a classical result of\nBrauer-Siegel\, or even ea
rlier result of Minkowski that the class number\n$|\\text{Cl}_K|$ of a num
ber field $K$ are always bounded by\n$\\text{Disc}(K)^{1/2+\\epsilon}$\, t
herefore we obtain a trivial bound\n$\\text{Disc}(K)^{1/2+\\epsilon}$ on $
|\\text{Cl}_K[\\ell]|$. We will talk\nabout results on this conjecture\, a
nd recent works on breaking the\ntrivial bound for $\\ell$-torsion of clas
s groups based on the work of\nEllenberg-Venkatesh.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Cass (Harvard)
DTSTART:20210302T000000Z
DTEND:20210302T005000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/24
DESCRIPTION:Title: A geometric construction of central elements in affine mod $p$ Hecke alg
ebras\nby Robert Cass (Harvard) as part of UCLA Number Theory Seminar\
n\n\nAbstract\nLet $G$ be a split connected reductive group over a local f
ield of positive characteristic. In the case of characteristic zero coeffi
cients\, Gaitsgory gave a geometric construction of central elements in th
e affine Hecke algebra of $G$ by applying a nearby cycles functor on a Bei
linson-Drinfeld affine Grassmannian. In this talk I will explain how to do
an analogous construction for the affine mod $p$ Hecke algebra of $G$. Ou
r techniques combine the geometry of Gaitsgory's construction (and simplif
ications due to Zhu) with perverse mod $p$ sheaves and tools from $F$-sing
ularities.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Iyengar (King's College London)
DTSTART:20210308T190000Z
DTEND:20210308T195000Z
DTSTAMP:20260422T225841Z
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 Seminar\n\n\
nAbstract\nIn this talk\, I will discuss a formulation of the Iwasawa main
conjecture in families over the extended eigencurve\, which is an extensi
on of the Coleman-Mazur eigencurve into characteristic $p$. This involves
constructing a family of $p$-adic $L$-functions\, a family of Galois repre
sentations\, and showing the characteristic ideal sheaves work in families
. I’ll give an overview and then give as much detail of the construction
as time permits.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pol van Hofton (King's College London)
DTSTART:20210329T170000Z
DTEND:20210329T175000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/26
DESCRIPTION:Title: Monodromy and irreducibility of Igusa varieties\nby Pol van Hofton (
King's College London) as part of UCLA Number Theory Seminar\n\n\nAbstract
\nIgusa varieties are smooth varieties in characteristic $p$ arising natur
ally as covers of certain subvarieties (central leaves) of Shimura varieti
es\, for example of the ordinary locus of the modular curve. The $\\ell$-a
dic cohomology of Igusa varieties acts as a bridge between the cohomology
of Rapoport-Zink spaces (local) and the cohomology of Shimura varieties (g
lobal)\, and it is therefore very interesting to study this cohomology. In
this talk I will discuss recent joint work with Luciena Xiao Xiao\, where
we compute the 0th cohomology group. This is equivalent to determining th
e irreducible components of Igusa varieties\, and our results generalise r
esults of Hida and Chai-Oort. Our strategy combines recent work of D’Add
ezio on monodromy of compatible local systems with a generalisation of a m
ethod of Hida\, and the Honda-Tate theory for Shimura varieties of Hodge t
ype of Kisin--Madapusi Pera--Shin.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sally Gilles (Imperial College London)
DTSTART:20210412T170000Z
DTEND:20210412T175000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/27
DESCRIPTION:Title: Syntomic cohomology and period morphisms\nby Sally Gilles (Imperial
College London) as part of UCLA Number Theory Seminar\n\n\nAbstract\nIn 20
17\, Colmez and Niziol proved a comparison theorem between arithmetic $p$-
adic nearby cycles and syntomic cohomology sheaves. To prove it\, they gav
e a local construction using $(\\phi\, \\Gamma)$-modules theory which allo
ws to reduce the period isomorphism to a comparison theorem between cohomo
logies of Lie algebras. I will explain the geometric version of this local
construction and how to globalize it to get a new period isomorphism. In
particular\, the explicit description of this new isomorphism can be used
to compare previous constructions of period morphisms and prove they are e
qual.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhixiang Wu (Université Paris-Saclay)
DTSTART:20210419T230000Z
DTEND:20210419T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/28
DESCRIPTION:Title: Companion forms and partially classical eigenvarieties\nby Zhixiang
Wu (Université Paris-Saclay) as part of UCLA Number Theory Seminar\n\n\nA
bstract\nIn general\, there exist $p$-adic automorphic forms of different
weights with the same associated $p$-adic Galois representation. In this t
alk\, I will report some result on the existence of companion forms for de
finite unitary groups when the Hodge-Tate weights are not regular\, genera
lizing the work of Breuil-Hellmann-Schraen in regular cases. One key ingre
dient of the proof is some partially classical properties of $p$-adic auto
morphic forms in the term of Emerton's Jacquet module for locally analytic
representations\, which will imply some partially de Rham properties of G
alois representations in finite slope cases with the help of Ding's partia
l eigenvarieties.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard)
DTSTART:20210426T230000Z
DTEND:20210426T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/29
DESCRIPTION:Title: Kolyvagin's conjecture and higher congruences of modular forms\nby N
aomi Sweeting (Harvard) as part of UCLA Number Theory Seminar\n\n\nAbstrac
t\nGiven an elliptic curve $E$\, Kolyvagin used CM points on modular curv
es 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 wi
ll report on recent work proving new cases of Kolyvagin's conjecture. The
proof builds on work of Wei Zhang\, who used congruences between modular f
orms to prove Kolyvagin's conjecture under some technical hypotheses. We r
emove many of these hypotheses by considering congruences modulo higher po
wers of $p$. The talk will explain the difficulties associated with higher
congruences of modular forms and how they can be overcome\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guido Kings (Universität Regensburg)
DTSTART:20210510T170000Z
DTEND:20210510T175000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/30
DESCRIPTION:Title: Equivariant Eisenstein classes\, critical values of Hecke $L$-functions
and $p$-adic interpolation\nby Guido Kings (Universität Regensburg) a
s part of UCLA Number Theory Seminar\n\n\nAbstract\nI report on joint work
with Johannes Sprang. Let $K$ be a CM field and\n$L/K$ be an extension of
degree $n$ and $\\chi$ be an algebraic critical Hecke\ncharacter of $L$.
Then we show that the $L$-value $L(\\chi\, 0)$ divided by\ncarefully norma
lized Shimura-Katz periods is integral and construct a\n$p$-adic $L$-funct
ion for $\\chi$. This generalizes results by Damerell\, Shimura and Katz f
or CM fields ($L = K$) and settles all open cases of algebraicity for crit
ical Hecke $L$-values.\n\nOur method relies on a detailed analysis of new
equivariant motivic Eisenstein classes and especially on the study of thei
r de Rham realizations and is completely different from the classical appr
oach by Shimura and Katz. The de Rham realization of these Eisenstein clas
ses\ncan be explicitly described in terms of Eisenstein-Kronecker series a
nd the equivariant setting is crucial to connect them with the $L$-functio
n of $\\chi$. An integral refinement of this construction leads directly t
o a geometric construction of a $p$-adic measure without any need to check
congruences for the Eisenstein series.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaoyu Zhang (Universität Duisburg-Essen)
DTSTART:20210503T170000Z
DTEND:20210503T175000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/31
DESCRIPTION:Title: CM points on compact orthogonal groups and theta lifts mod $p$\nby X
iaoyu Zhang (Universität Duisburg-Essen) as part of UCLA Number Theory Se
minar\n\n\nAbstract\nCM points on Shimura varieties are very useful in the
study of arithmetic properties of automorphic forms\, $L$-functions\, etc
.. C. Cornut and V. Vatsal proved certain dynamical properties of these po
ints on quaternions and then deduce Mazur's conjectures as well as study t
he non-vanishing of Rankin-Selberg $L$-values. In this talk we try to foll
ow the strategy of Cornut and Vatsal and generalise their result to certai
n compact orthogonal groups and as an application\, we study the non-vanis
hing problem of theta lifts mod $p$ from orthogonal group to symplectic gr
oup.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Hough (Stony Brook)
DTSTART:20211004T220000Z
DTEND:20211004T225000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/32
DESCRIPTION:Title: Recent developments in orbit counting methods\nby Robert Hough (Ston
y Brook) as part of UCLA Number Theory Seminar\n\n\nAbstract\nBhargava pio
neered methods from the geometry of numbers to count integral orbits in re
presentation spaces ordered by invariants. I will discuss recent analytic
techniques in development to strengthen the methods\, including spectral
expansion of the underlying homogeneous spaces\, classification of local o
rbits and their finite Fourier transforms\, and subconvex estimates for th
e enumerating zeta functions. In particular\, we have obtained a strong a
nswer to a question of Bhargava and Gross at a conference at the American
Institute of Math explaining a barrier to equidistribution in the shape of
cubic fields by obtaining poles and residues in the zeta function enumera
ting the Weyl sums in the Eisenstein spectrum. Joint work with Eun Hye Le
e.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Florea (UC Irvine)
DTSTART:20211018T220000Z
DTEND:20211018T225000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/33
DESCRIPTION:Title: The Ratios Conjecture over function fields\nby Alexandra Florea (UC
Irvine) as part of UCLA Number Theory Seminar\n\n\nAbstract\nI will talk a
bout some recent joint work with H. Bui and J. Keating where we study the
Ratios Conjecture for the family of quadratic L-functions over function fi
elds. I will also discuss the closely related problem of obtaining upper b
ounds for negative moments of L-functions\, which allows us to obtain part
ial results towards the Ratios Conjecture in the case of one over one\, tw
o over two and three over three L-functions.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Horawa (Michigan)
DTSTART:20211011T220000Z
DTEND:20211011T225000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/34
DESCRIPTION:Title: Motivic action on coherent cohomology of Hilbert modular varieties\n
by Alex Horawa (Michigan) as part of UCLA Number Theory Seminar\n\n\nAbstr
act\nA surprising property of cohomology of locally symmetric spaces is th
at Hecke operators can act on multiple cohomological degrees with the same
eigenvalues. We will discuss this phenomenon for coherent cohomology of l
ine bundles on modular curves and\, more generally\, Hilbert modular varie
ties. We propose an arithmetic explanation: a hidden degree-shifting actio
n of a certain motivic cohomology group (the Stark unit group). This exten
ds the conjectures of Venkatesh\, Prasanna\, and Harris to Hilbert modular
varieties.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashay Burungale (Caltech)
DTSTART:20211025T220000Z
DTEND:20211025T225000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/35
DESCRIPTION:Title: p-converse to a theorem of Gross-Zagier and Kolyvagin - dihedral primes<
/a>\nby Ashay Burungale (Caltech) as part of UCLA Number Theory Seminar\n\
nLecture held in Math Science Building 5203.\n\nAbstract\nSuch a p-convers
e will be outlined (joint with Chris Skinner).\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anwesh Ray (Univ of British Columbia)
DTSTART:20211101T220000Z
DTEND:20211101T225000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/36
DESCRIPTION:Title: Iwasawa theory and congruences for the symmetric square of a modular for
m\nby Anwesh Ray (Univ of British Columbia) as part of UCLA Number The
ory Seminar\n\n\nAbstract\nI will report on joint work with R. Sujatha and
V. Vatsal. Two\n$p$-ordinary Hecke-eigenforms are are congruent at a prim
e $\\varpi|p$ if\nall but finitely many of their Fourier coefficients are
congruent modulo\n$\\varpi$. R. Greenberg and V. Vatsal showed in 2000 tha
t the\nIwasawa-invariants of congruent modular forms are related. As a res
ult\, if\n$\\mu$-invariant vanishes and the main conjecture holds for a gi
ven\nHecke-eigenform\, then the same is true for a congruent Hecke-eigenfo
rm.\nThis involves studying the behavior of Selmer groups and p-adic L-fun
ctions\nwith respect to congruences. We generalize these results to symmet
ric\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 terms of th
e Petersson inner product of $f$\nwith a nearly holomorphic function. This
function is expressed as the\nproduct of a theta function and an Eisentei
n series. The ordinary\nholomorphic projection of this function is shown t
o have nice properties.\nThe Petersson inner product is modified and relat
ed to an abstractly\ndefined algebraic pairing due to Hida\, and the two p
airing are related up\nto a "canonical period". Under further hypotheses\,
it is shown that this\ncanonical period is suitably well behaved. For thi
s\, we assume a certain\nversion of Ihara's lemma\, which is known in cert
ain cases.\n\n With these preparations\, we are able to show that normaliz
ed L-values for\nthe symmetric square behave well with respect to congruen
ce\, and hence\, the\np-adic L-functions too. It follows that the analytic
Iwasawa invariants for congruent Hecke-eigencuspforms are related. Such r
esults for the algebraic Iwasawa invariants follow from work of R. Greenbe
rg and V. Vatsal. Just as in the classicial case\, the results have implic
ations to the main conjecture. If time permits\, we will introduce the rol
e of the fine-Selmer\ngroup and discuss a condition for the vanishing on t
he $\\mu$-invariant that\ncan be stated purely in terms of the residual re
presentation.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brandon Alberts (UC San Diego)
DTSTART:20211129T230000Z
DTEND:20211129T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/37
DESCRIPTION:Title: Refining Malle's Conjecture for Inductive Counting Methods\nby Brand
on Alberts (UC San Diego) as part of UCLA Number Theory Seminar\n\nLecture
held in Math Science Building 5203.\n\nAbstract\nMalle's conjecture predi
cts the asymptotic growth rate of the number of G-extensions F/K of a numb
er field K with absolute discriminant bounded above by X\, where X tends t
owards infinity. I will discuss a joint project with Robert Lemke Oliver\,
Jiuya Wang\, and Melanie Matchett Wood to approach this conjecture induct
ively by first restricting to G-extensions F/K containing a fixed intermed
iate extension L/K\, then taking a sum over choices of intermediate extens
ions. A fundamental concept in this talk will be the related question of f
inding the distribution of elements of the first Galois cohomology group\,
$H^1(K\,T)$. In particular\, I will address a joint paper with Evan O'Dor
ney using harmonic analysis to study $H^1(K\,T)$.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yong Suk Moon (Univ of Arizona)
DTSTART:20211108T230000Z
DTEND:20211108T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/38
DESCRIPTION:Title: Prismatic crystals and crystalline representations in the relative case<
/a>\nby Yong Suk Moon (Univ of Arizona) as part of UCLA Number Theory Semi
nar\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-Scholze r
ecently proved that the category of prismatic F-crystals on the absolute p
rismatic site over O_K is equivalent to the category of lattices of crysta
lline representations of G_K. We study an analogous situation in the relat
ive 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 certain comp
leted F-crystals on the absolute prismatic site over R to the category of
crystalline Z_p-local systems on the generic fiber of Spf R. Furthermore\,
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://researchseminars.org/talk/UCLA_NTS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gilyoung Cheong (UC Irvine)
DTSTART:20211115T230000Z
DTEND:20211115T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/39
DESCRIPTION:Title: Counting square-free numbers in arithmetic geometry\nby Gilyoung Che
ong (UC Irvine) as part of UCLA Number Theory Seminar\n\n\nAbstract\nWe de
lve into an innocuous question about counting\n"square-free numbers" in va
rious forms\, following the philosophy of\nWeil's three columns.\n\nExampl
e 1. We count square-free integers.\n\nExample 2. We count square-free pol
ynomials of a fixed degree over a\nfinite field.\n\nExample 3. We compute
the Betti numbers of the space of square-free\npolynomials of a fixed degr
ee over complex numbers by quoting a theorem\nof Arnol'd in topology.\n\nB
y viewing Example 3 as counting square-free 0-cycles on the affine line\no
ver complex numbers\, we add one more example to this list\, using our\nma
in result.\n\nExample 4. We compute the Betti numbers of the space of squa
re-free\n0-cycles of a fixed degree on a punctured elliptic curve over com
plex\nnumbers.\n\nWe briefly explain how Examples 3 and 4 can be obtained
by showing that\nthe mixed Hodge structure of the i-th singular cohomology
group with\nrational coefficients is pure of some weight different from i
. This is\njoint work with Yifeng Huang.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UC San Diego)
DTSTART:20211122T230000Z
DTEND:20211122T235000Z
DTSTAMP:20260422T225841Z
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 Science Buil
ding 5203.\n\nAbstract\nI will report on joint work-in-progress with Spenc
er Leslie where we define an analogue of the Cohen-Zagier Eisenstein serie
s to the exceptional group G_2. Recall that the Cohen-Zagier Eisenstein se
ries is a weight 3/2 modular form whose Fourier coefficients see the class
numbers of imaginary quadratic fields. We define a particular modular for
m of weight 1/2 on G_2\, and prove that its Fourier coefficients see the 2
-torsion in the narrow class groups of totally real cubic fields. In parti
cular: 1) we define a notion of modular forms of half-integral weight on c
ertain exceptional groups\, 2) we prove that these modular forms have a ni
ce theory of Fourier coefficients\, and 3) we partially compute the Fourie
r coefficients of a particular nice example on G_2.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Lawrence (UCLA)
DTSTART:20220110T230000Z
DTEND:20220110T235000Z
DTSTAMP:20260422T225841Z
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\nInter
esting moduli spaces don't have many integral points. More precisely\, if
X is a variety over a number field\, admitting a variation of Hodge struct
ure whose associate period map is injective\, then the number of S-integra
l points on X of height at most H grows more slowly than H^{\\epsilon}\, f
or any positive \\epsilon. This is a sort of weak generalization of the Sh
afarevich conjecture\; it is a consequence of a point-counting theorem of
Broberg\, and the largeness of the fundamental group of X. Joint with Elle
nberg and Venkatesh.\n\nhttps://arxiv.org/abs/2109.01043\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Harris (Columbia)
DTSTART:20220112T230000Z
DTEND:20220112T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/42
DESCRIPTION:Title: L-functions and periods of automorphic forms\nby Michael Harris (Col
umbia) as part of UCLA Number Theory Seminar\n\n\nAbstract\nThis is a repo
rt on recent work with Grobner and Lin on the critical values of\nRankin-S
elberg L-functions of GL(n)xGL(n-1) over CM fields. By reinterpreting th
ese critical values in terms of automorphic periods of holomorphic automor
phic forms on unitary groups\, we show that the automorphic periods of hol
omorphic forms can be factored as products of coherent cohomological forms
\, compatibly with a motivic factorization predicted by the Tate conjectur
e. All of these results are conditional on a conjecture on non-vanishing o
f twists of automorphic L-functions of GL(n) by anticyclotomic characters
of finite order\, and are stated under a certain regularity condition.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hang Xue (Univ. of Arizona)
DTSTART:20220124T230000Z
DTEND:20220124T235000Z
DTSTAMP:20260422T225841Z
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\nAbs
tract\nI explain a very simple proof of the local Gan--Gross--Prasad conje
cture for real unitary groups. I also explain meromorphic continuation of
the explicit tempered intertwining map based on a similar idea.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UC Santa Barbara)
DTSTART:20220131T230000Z
DTEND:20220131T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/44
DESCRIPTION:Title: Selmer classes on CM elliptic curves of rank 2\nby Francesc Castella
(UC Santa Barbara) as part of UCLA Number Theory Seminar\n\nLecture 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 pioneeri
ng 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-converse 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 infinite orde
r. In this talk\, I'll report on the proof of an analogue of Skinner's res
ult in the rank 2 case\, in which Heegner points are replaced by certain g
eneralized Kato classes introduced by Darmon-Rotger. For E without CM\, su
ch an analogue was obtained in an earlier work with M.-L. Hsieh\, and in t
his talk I'll focus on the CM case\, whose proof uses a different set of i
deas.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Dunn (Caltech)
DTSTART:20220207T230000Z
DTEND:20220207T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/45
DESCRIPTION:Title: Bias in cubic Gauss sums: Patterson's conjecture\nby Alex Dunn (Calt
ech) as part of UCLA Number Theory Seminar\n\nLecture held in Math Science
5118.\n\nAbstract\nWe prove\, in this joint work with Maksym Radziwill\,
a 1978 conjecture of S. Patterson (conditional on the Generalised Riemann
hypothesis) concerning the bias of cubic Gauss sums. This explains a well-
known numerical bias in the distribution of cubic Gauss sums first observe
d by Kummer in 1846.\n\nThere are two important byproducts of our proof. T
he first is an explicit level aspect Voronoi summation formula for cubic G
auss sums\, extending computations of Patterson and Yoshimoto. Secondly\,
we show that Heath-Brown's cubic large sieve is sharp under GRH. This disp
roves the popular belief that the cubic large sieve can be improved.\n\nAn
important ingredient in our proof is a dispersion estimate for cubic Gaus
s sums. It can be interpreted as a cubic large sieve with correction by a
non-trivial asymptotic main term. This estimate relies on the Generalised
Riemann Hypothesis\, and is one of the fundamental reasons why our result
is conditional.\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mingjie Chen (UC San Diego)
DTSTART:20220214T230000Z
DTEND:20220214T235000Z
DTSTAMP:20260422T225841Z
UID:UCLA_NTS/46
DESCRIPTION:Title: Orienteering with one endomorphism\nby Mingjie Chen (UC San Diego) a
s part of UCLA Number Theory Seminar\n\n\nAbstract\nSupersingular isogeny-
based cryptosystems are strong contenders for post-quantum cryptography st
andardization. Such cryptosystems rely on the hardness of path-finding on
supersingular isogeny graphs. The path-finding problem is known to reduce
to the endomorphism ring problem. Can path-finding be reduced to knowing j
ust one endomorphism? In this talk\, we give explicit classical and quantu
m algorithms for path-finding to an initial curve using the knowledge of o
ne endomorphism. An endomorphism gives an orientation of a supersingular e
lliptic curve. We use the theory of oriented supersingular isogeny graphs
and algorithms for taking ascending/descending/horizontal steps on such gr
aphs.\n\npaper link: https://arxiv.org/pdf/2201.11079.pdf\n
LOCATION:https://researchseminars.org/talk/UCLA_NTS/46/
END:VEVENT
END:VCALENDAR