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BEGIN:VEVENT
SUMMARY:Jishnu Ray (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20200423T210000Z
DTEND;VALUE=DATE-TIME:20200423T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/1
DESCRIPTION:Title: Conjectures in Iwasawa Theory of Selmer groups and Iwasawa Algebras\n
by Jishnu Ray (University of British Columbia) as part of UCSD number theo
ry seminar\n\n\nAbstract\nThe Iwasawa Theory of Selmer groups provides a n
atural way for p-adic approach to the celebrated Birch and Swinnerton Dyer
conjecture. Over a non-commutative p-adic Lie extension\, the (dual) Selm
er group becomes a module over a non-commutative Iwasawa algebra and its s
tructure can be understood by analyzing “(left) reflexive ideals” in t
he Iwasawa algebra. In this talk\, we will start by recalling several exis
ting conjectures in Iwasawa Theory and then we will give an explicit ring-
theoretic presentation\, by generators and relations\, of such Iwasawa alg
ebras and sketch its implications in understanding the (two-sides) reflexi
ve ideals. Generalizing Clozel’s work for SL(2)\, we will also show that
such an explicit presentation of Iwasawa algebras can be obtained for a m
uch wider class of p-adic Lie groups viz. uniform pro-p groups and the pro
-p Iwahori of GL(n\,Z_p). Further\, if time permits\, I will also sketch s
ome of my recent Iwasawa theoretic results joint with Sujatha Ramdorai.\n\
npretalk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jize Yu (California Institute of Technology)
DTSTART;VALUE=DATE-TIME:20200430T210000Z
DTEND;VALUE=DATE-TIME:20200430T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/2
DESCRIPTION:Title: The integral geometric Satake equivalence in mixed characteristic\nby
Jize Yu (California Institute of Technology) as part of UCSD number theor
y seminar\n\nLecture held in APM 7321.\n\nAbstract\nThe geometric Satake e
quivalence establishes a link between two monoidal categories: the categor
y of perverse sheaves on the local Hecke stack and the category of finitel
y generated representations of the Langlands dual group. It has many impor
tant applications in the study of the geometric Langlands program and numb
er theory. In this talk\, I will discuss the integral coefficient geometri
c Satake equivalence in the mixed characteristic setting. It generalizes t
he previous results of Lusztig\, Ginzburg\, Mirkovic-Vilonen\, and Zhu. Ti
me permitting\, I will discuss an application of this result in constructi
ng a Jacquet-Langlands transfer.\n\nThere will be a pretalk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Wang-Erickson (University of Pittsburgh)
DTSTART;VALUE=DATE-TIME:20200507T210000Z
DTEND;VALUE=DATE-TIME:20200507T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/3
DESCRIPTION:Title: The Eisenstein ideal with squarefree level\nby Carl Wang-Erickson (Un
iversity of Pittsburgh) as part of UCSD number theory seminar\n\nLecture h
eld in APM 7321.\n\nAbstract\nIn his landmark paper "Modular forms and the
Eisenstein ideal\," Mazur studied congruences modulo a prime p between th
e Hecke eigenvalues of an Eisenstein series and the Hecke eigenvalues of c
usp forms\, assuming these modular forms have weight 2 and prime level N.
He asked about generalizations to squarefree levels N. I will present some
work on such generalizations\, which is joint with Preston Wake and Cathe
rine Hsu.\n\nThere will be a pretalk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Thorne (Cambridge University)
DTSTART;VALUE=DATE-TIME:20200521T210000Z
DTEND;VALUE=DATE-TIME:20200521T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/5
DESCRIPTION:Title: Symmetric power functoriality for holomorphic modular forms\nby Jack
Thorne (Cambridge University) as part of UCSD number theory seminar\n\nLec
ture held in APM 7321.\n\nAbstract\nLanglands’s functoriality conjecture
s predict the existence of “liftings” of automorphic representations a
long morphisms of L-groups. A basic case of interest comes from the irredu
cible algebraic representations of GL(2)\, thought of as the L-group of th
e reductive group GL(2) over Q. I will discuss the proof\, joint with Jame
s Newton\, of the existence of the corresponding functorial liftings for
a broad class of holomorphic modular forms\, including Ramanujan’s Delta
function.\n\nThere will be a pre-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Fuchs (University of California\, Davis)
DTSTART;VALUE=DATE-TIME:20200528T210000Z
DTEND;VALUE=DATE-TIME:20200528T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/6
DESCRIPTION:Title: Prime components in integral circle packings\nby Elena Fuchs (Univers
ity of California\, Davis) as part of UCSD number theory seminar\n\nLectur
e held in APM 7321.\n\nAbstract\nCircle packings in which all circles have
integer curvature\, particularly Apollonian circle packings\, have in the
last decade become objects of great interest in number theory. In this ta
lk\, we explore some of their most fascinating arithmetic features\, from
local to global properties to prime components in the packings\, going fro
m theorems\, to widely believed conjectures\, to wild guesses as to what m
ight be true.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niccolo Ronchetti (University of California\, Los Angeles)
DTSTART;VALUE=DATE-TIME:20200604T210000Z
DTEND;VALUE=DATE-TIME:20200604T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/7
DESCRIPTION:Title: A derived Hecke action on the ordinary Hida tower\nby Niccolo Ronchet
ti (University of California\, Los Angeles) as part of UCSD number theory
seminar\n\nLecture held in APM 7321.\n\nAbstract\nWhen studying the cohomo
logy of Shimura varieties and arithmetic manifolds\, one encounters the fo
llowing phenomenon: the same Hecke eigensystem shows up in multiple degree
s around the middle dimension\, and its multiplicities in these degrees re
sembles that of an exterior algebra.\n\nIn a series of recent papers\, Ven
katesh and his collaborators provide an explanation: they construct graded
objects having a graded action on the cohomology\, and show that under go
od circumstances this action factors through that of an explicit exterior
algebra\, which in turn acts faithfully and generate the entire Hecke eige
nspace.\n\nIn this talk\, we discuss joint work with Khare where we invest
igate the $p=p$ situation (as opposed to the $l \\neq p$ situation\, which
is the main object of study of Venkatesh’s Derived Hecke Algebra paper)
: we construct a degree-raising action on the cohomology of the ordinary H
ida tower and show that (under some technical assumptions)\, this action g
enerates the full Hecke eigenspace under its lowest nonzero degree. Then\,
we bring Galois representations into the picture\, and show that the deri
ved Hecke action constructed before is in fact related to the action of a
certain dual Selmer group.\n\nThere will be a pre-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Pellarin (U. Jean Monnet\, Saint-Etienne\, France)
DTSTART;VALUE=DATE-TIME:20200514T170000Z
DTEND;VALUE=DATE-TIME:20200514T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/8
DESCRIPTION:Title: On Drinfeld modular forms in Tate algebras\nby Federico Pellarin (U.
Jean Monnet\, Saint-Etienne\, France) as part of UCSD number theory semina
r\n\nLecture held in APM 7321.\n\nAbstract\nIn this talk we will describe
some recent works on Drinfeld modular forms with values in Tate algebras (
in 'equal positive characteristic'). In particular\, we will discuss some
remarkable identities (proved or conjectural) for Eisenstein and Poincaré
series\, and the problem of analytically interpolate families of Drinfeld
modular forms for congruence subgroups at the infinity place.\n\nThe pre-
talk will begin 30 minutes prior (09:30 local time).\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xin Tong (University of California\, San Diego)
DTSTART;VALUE=DATE-TIME:20200514T210000Z
DTEND;VALUE=DATE-TIME:20200514T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/10
DESCRIPTION:Title: Towards a Hodge-Iwasawa theory\nby Xin Tong (University of Californi
a\, San Diego) as part of UCSD number theory seminar\n\n\nAbstract\nWith t
he motivation of generalizing the corresponding geometrization of Tamagawa
-Iwasawa theory after Kedlaya-Pottharst\, and with motivation of establish
ing the corresponding equivariant version of the relative p-adic Hodge the
ory after Kedlaya-Liu aiming at the deformation of representations of prof
inite fundamental groups and the family of étale local systems\, we initi
ate the corresponding Hodge-Iwasawa theory with deep point of view and phi
losophy in mind from early work of Kato and Fukaya-Kato. In this talk\, we
are going to discuss some foundational results on the Hodge-Iwasawa modul
es and Hodge-Iwasawa sheaves\, as well as some interesting investigation t
owards the goal in our mind\, which are taken from our first paper in this
series project.\n\nThere will be a pre-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Organizational meeting (UCSD)
DTSTART;VALUE=DATE-TIME:20201001T210000Z
DTEND;VALUE=DATE-TIME:20201001T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/11
DESCRIPTION:Title: Organizational meeting\nby Organizational meeting (UCSD) as part of
UCSD number theory seminar\n\nLecture held in normally APM 7321\, currentl
y online.\n\nAbstract\nThis is an organizational meeting for the remainder
of the term. The seminar itself will begin one week later.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UCSD)
DTSTART;VALUE=DATE-TIME:20201008T210000Z
DTEND;VALUE=DATE-TIME:20201008T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/12
DESCRIPTION:Title: Singular modular forms on quaternionic E_8\nby Aaron Pollack (UCSD)
as part of UCSD number theory seminar\n\nLecture held in normally APM 7321
\, currently online.\n\nAbstract\nThe exceptional group $E_{7\,3}$ has a s
ymmetric space with Hermitian tube structure. On it\, Henry Kim wrote dow
n low weight holomorphic modular forms that are "singular" in the sense th
at their Fourier expansion has many terms equal to zero. The symmetric sp
ace associated to the exceptional group $E_{8\,4}$ does not have a Hermiti
an structure\, but it has what might be the next best thing: a quaternioni
c structure and associated "modular forms". I will explain the constructio
n of singular modular forms on $E_{8\,4}$\, and the proof that these speci
al modular forms have rational Fourier expansions\, in a precise sense. T
his builds off of work of Wee Teck Gan and uses key input from Gordan Savi
n.\n\npre-talk at 1:30pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Mundy (Columbia)
DTSTART;VALUE=DATE-TIME:20201105T220000Z
DTEND;VALUE=DATE-TIME:20201105T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/13
DESCRIPTION:Title: Archimedean components of Eisenstein series and CAP forms for $G_2$\
nby Samuel Mundy (Columbia) as part of UCSD number theory seminar\n\nLectu
re held in normally APM 7321\, currently online.\n\nAbstract\nI will talk
about some recent work determining the archimedean components of certain E
isenstein series and CAP forms induced from the long root parabolic of $G_
2$. I will also discuss how these results are being used in some work in p
rogress on producing nonzero classes in symmetric cube Selmer groups.\n\np
re-talk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brandon Alberts (UCSD)
DTSTART;VALUE=DATE-TIME:20201029T210000Z
DTEND;VALUE=DATE-TIME:20201029T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/14
DESCRIPTION:Title: Modeling Malle's Conjecture with Random Groups\nby Brandon Alberts (
UCSD) as part of UCSD number theory seminar\n\nLecture held in normally AP
M 7321\, currently online.\n\nAbstract\nWe construct a random group with a
local structure that models the behavior of the absolute Galois group ${\
\rm Gal}(\\overline{K}/K)$\, and prove that this random group satisfies Ma
lle's conjecture for counting number fields ordered by discriminant with p
robability 1. This work is motivated by the use of random groups to model
class group statistics in families of number fields (and generalizations).
We take care to address the known counter-examples to Malle's conjecture
and how these may be incorporated into the random group.\n\npre-talk at 1:
30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Upton (UCSD)
DTSTART;VALUE=DATE-TIME:20201112T220000Z
DTEND;VALUE=DATE-TIME:20201112T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/15
DESCRIPTION:Title: Newton Slopes in $\\mathbb{Z}_p$-Towers of Curves\nby James Upton (U
CSD) as part of UCSD number theory seminar\n\nLecture held in normally APM
7321\, currently online.\n\nAbstract\nLet $X/\\mathbb{F}_q$ be a smooth a
ffine curve over a finite field of characteristic $p > 2$. In this talk we
discuss the $p$-adic variation of zeta functions $Z(X_n\,s)$ in a pro-cov
ering $X_\\infty:\\cdots \\to X_1 \\to X_0 = X$ with total Galois group $\
\mathbb{Z}_p$. For certain ``monodromy stable'' coverings over an ordinary
curve $X$\, we prove that the $q$-adic Newton slopes of $Z(X_n\,s)/Z(X\,s
)$ approach a uniform distribution in the interval $[0\,1]$\, confirming a
conjecture of Daqing Wan. We also prove a ``Riemann hypothesis'' for a fa
mily of Galois representations associated to $X_\\infty/X$\, analogous to
the Riemann hypothesis for equicharacteristic $L$-series as posed by Davi
d Goss. This is joint work with Joe Kramer-Miller.\n\npre-talk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yifeng Liu (Yale University)
DTSTART;VALUE=DATE-TIME:20201120T000000Z
DTEND;VALUE=DATE-TIME:20201120T010000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/16
DESCRIPTION:Title: Beilinson-Bloch conjecture and arithmetic inner product formula\nby
Yifeng Liu (Yale University) as part of UCSD number theory seminar\n\nLect
ure held in normally APM 7321\, currently online.\n\nAbstract\nIn this tal
k\, we study the Chow group of the motive associated to a tempered global
L-packet \\pi of unitary groups of even rank with respect to a CM extensio
n\, whose global root number is -1. We show that\, under some restrictions
on the ramification of \\pi\, if the central derivative L'(1/2\,\\pi) is
nonvanishing\, then the \\pi-nearly isotypic localization of the Chow grou
p of a certain unitary Shimura variety over its reflex field does not vani
sh. This proves part of the Beilinson--Bloch conjecture for Chow groups an
d L-functions. Moreover\, assuming the modularity of Kudla's generating fu
nctions of special cycles\, we explicitly construct elements in a certain
\\pi-nearly isotypic subspace of the Chow group by arithmetic theta liftin
g\, and compute their heights in terms of the central derivative L'(1/2\,\
\pi) and local doubling zeta integrals. This is a joint work with Chao Li.
\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Mornev (ETHZ)
DTSTART;VALUE=DATE-TIME:20201203T180000Z
DTEND;VALUE=DATE-TIME:20201203T190000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/17
DESCRIPTION:Title: Local monodromy of Drinfeld modules\nby Maxim Mornev (ETHZ) as part
of UCSD number theory seminar\n\nLecture held in normally APM 7321\, curre
ntly online.\n\nAbstract\nThe theory of Drinfeld modules is remarkably sim
ilar to the theory of abelian varieties\, but their local monodromy behave
s differently and is poorly understood. In this talk I will present a rese
arch program which aims to fully describe this monodromy. The cornerstone
of this program is a "z-adic" variant of Grothendieck's l-adic monodromy t
heorem.\n\nThe talk is aimed at a general audience of number theorists and
arithmetic geometers. No special knowledge of monodromy theory or Drinfel
d modules is assumed.\n\nThere will be a pre-talk introducing the theory o
f t-motives.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristian Popescu (UCSD)
DTSTART;VALUE=DATE-TIME:20201015T210000Z
DTEND;VALUE=DATE-TIME:20201015T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/18
DESCRIPTION:Title: An equivariant Tamagawa number formula for Drinfeld modules and beyond\nby Cristian Popescu (UCSD) as part of UCSD number theory seminar\n\nLe
cture held in normally APM 7321\, currently online.\n\nAbstract\nI will pr
esent a vast generalization of Taelman's 2012 celebrated class-number form
ula for Drinfeld modules to the setting of (rigid analytic) L-functions of
Drinfeld module motives with Galois equivariant coefficients. I will disc
uss applications and potential extensions of this formula to the category
of t-modules and t-motives. This is based on joint work with Ferrara\, Gre
en and Higgins\, and a result of meetings in the UCSD Drinfeld Module Semi
nar.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Van Koughnett (Purdue)
DTSTART;VALUE=DATE-TIME:20201022T210000Z
DTEND;VALUE=DATE-TIME:20201022T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/19
DESCRIPTION:Title: Topological modular forms for number theorists\nby Paul Van Koughnet
t (Purdue) as part of UCSD number theory seminar\n\nLecture held in normal
ly APM 7321\, currently online.\n\nAbstract\nThis will be a mainly exposit
ory talk about some recent applications of number theory to topology. The
crux of these applications is the construction of a cohomology theory call
ed topological modular forms (TMF) out of the moduli of elliptic curves. I
'll explain what TMF is\, what we have been doing with it\, and what we'd
still like to know\; I'll also discuss more recent attempts to extend the
theory using level structures\, higher-dimensional abelian varieties\, and
K3 surfaces. Time permitting\, I'll talk about my work with Dominic Culve
r on some partial number-theoretic interpretations of TMF co-operations.\n
\nI'll give a pre-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bao Le Hung (Northwestern University)
DTSTART;VALUE=DATE-TIME:20201210T220000Z
DTEND;VALUE=DATE-TIME:20201210T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/20
DESCRIPTION:Title: Moduli of Fontaine-Laffaille modules and mod p local-global compatibilit
y.\nby Bao Le Hung (Northwestern University) as part of UCSD number th
eory seminar\n\nLecture held in normally APM 7321\, currently online.\n\nA
bstract\nThe mod p cohomology of locally symmetric spaces for definite uni
tary groups at infinite level is expected to realize the mod p local Langl
ands correspondence for GL_n. In particular\, one expects the (component a
t p) of the associated Galois representation to be determined by cohomolog
y as a smooth representation. I will describe how one can establish this e
xpectation in many cases when the local Galois representation is Fontaine-
Laffaille.\nThis is joint work with D. Le\, S. Morra\, C. Park and Z. Qian
.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Lam (Harvard University)
DTSTART;VALUE=DATE-TIME:20210107T220000Z
DTEND;VALUE=DATE-TIME:20210107T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/21
DESCRIPTION:Title: Calabi-Yau varieties and Shimura varieties\nby Joshua Lam (Harvard U
niversity) as part of UCSD number theory seminar\n\nLecture held in normal
ly APM 7321\, currently online.\n\nAbstract\nI will discuss the Attractor
Conjecture for Calabi-Yau varieties\, which was formulated by Moore in the
nineties\, highlighting the difference between Calabi-Yau varieties with
and without Shimura moduli. In the Shimura case\, I show that the conjectu
re holds and gives rise to an explicit parametrization of CM points on cer
tain Shimura varieties\; in the case without Shimura moduli\, I’ll prese
nt counterexamples to the conjecture using unlikely intersection theory. P
art of this is joint work with Arnav Tripathy.\n\nThere will be a 30 minut
e pre-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aranya Lahiri (Indiana University)
DTSTART;VALUE=DATE-TIME:20210114T220000Z
DTEND;VALUE=DATE-TIME:20210114T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/22
DESCRIPTION:Title: Resolutions of locally analytic principal series representations of GL_2
(F)\nby Aranya Lahiri (Indiana University) as part of UCSD number theo
ry seminar\n\nLecture held in normally APM 7321\, currently online.\n\nAbs
tract\nLocally analytic representations of $p$-adic analytic groups have p
layed a crucial role in many areas of arithmetic and representation theory
(including in $p$-adic local Langlands program) since their introduction
by Schneider and Teitelbaum. In this talk we will briefly review some asp
ects of the theory of locally analytic representations. Then\, for a loca
lly analytic representation $V$ of $GL_2(F)$ we will construct a coefficie
nt system attached to the Bruhat-Tits tree of $Gl_2(F)$. Finally we will u
se this coefficient system to construct a resolution for locally analytic
principal series of $GL_2(F)$.\n\npre-talk at 1:30. I will discuss basics
and some key examples of locally analytic representations in the pre-talk.
\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard University)
DTSTART;VALUE=DATE-TIME:20210204T220000Z
DTEND;VALUE=DATE-TIME:20210204T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/23
DESCRIPTION:Title: Kolyvagin's conjecture and higher congruences of modular forms\nby N
aomi Sweeting (Harvard University) as part of UCSD number theory seminar\n
\nLecture held in normally APM 7321\, currently online.\n\nAbstract\nGiven
an elliptic curve E\, Kolyvagin used CM points on modular curves to cons
truct a system of classes valued in the Galois cohomology of the torsion p
oints of E. Under the conjecture that not all of these classes vanish\, h
e gave a description for the Selmer group of E. This talk will report on
recent work proving new cases of Kolyvagin's conjecture. The methods foll
ow in the footsteps of Wei Zhang\, who used congruences between modular fo
rms to prove Kolyvagin's conjecture under some technical hypotheses. We re
move 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. I will also pro
vide an introduction to the conjecture and its consequences\, including a
'converse theorem': algebraic rank one implies analytic rank one.\n\npre-t
alk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kwun Angus Chung (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210121T220000Z
DTEND;VALUE=DATE-TIME:20210121T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/24
DESCRIPTION:Title: $v$-adic convergence for exp and log in function fields and applications
to $v$-adic $L$-values\nby Kwun Angus Chung (University of Michigan)
as part of UCSD number theory seminar\n\nLecture held in normally APM 7321
\, currently online.\n\nAbstract\nClassically over the rational numbers\,
the exponential and logarithm series converge $p$-adically within some ope
n disc of $\\mathbb{C}_p$. For function fields\, exponential and logarithm
series arise naturally from Drinfeld modules\, which are objects construc
ted by Drinfeld in his thesis to prove the Langlands conjecture for $\\mat
hrm{GL}_2$ over function fields. For a "finite place" $v$ on such a curve\
, one can ask if the exp and log possess similar $v$-adic convergence prop
erties. For the most basic case\, namely that of the Carlitz module over $
\\mathbb{F}_q[T]$\, this question has been long understood. In this talk\,
we will show the $v$-adic convergence for Drinfeld-(Hayes) modules on ell
iptic curves and a certain class of hyperelliptic curves. As an applicatio
n\, we are then able to obtain a formula for the $v$-adic $L$-value $L_v(1
\,\\Psi)$ for characters in these cases\, analogous to Leopoldt's formula
in the number field case.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Iyengar (King's College\, London)
DTSTART;VALUE=DATE-TIME:20210128T220000Z
DTEND;VALUE=DATE-TIME:20210128T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/25
DESCRIPTION:Title: The Iwasawa Main Conjecture over the Extended Eigencurve\nby Ashwin
Iyengar (King's College\, London) as part of UCSD number theory seminar\n\
nLecture held in normally APM 7321\, currently online.\n\nAbstract\nI will
give a brief historical motivation for the Iwasawa main conjecture\, and
then I will talk about a construction of a $p$-adic $L$-function in famili
es over the extended eigencurve\, and how to formulate a two-variable Iwas
awa main conjecture. If time permits\, I will state some open questions ab
out this family of functions.\n\nI will give a pre-talk beforehand at 1:30
.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano Lopez (University of Utah)
DTSTART;VALUE=DATE-TIME:20210211T220000Z
DTEND;VALUE=DATE-TIME:20210211T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/26
DESCRIPTION:Title: Counting elliptic curves with prescribed torsion over imaginary quadrati
c fields\nby Allechar Serrano Lopez (University of Utah) as part of UC
SD number theory seminar\n\nLecture held in normally APM 7321\, currently
online.\n\nAbstract\nA generalization of Mazur's theorem states that there
are 26 possibilities for the torsion subgroup of an elliptic curve over a
quadratic extension of $\\mathbb{Q}$. If $G$ is one of these groups\, we
count the number of elliptic curves of bounded naive height whose torsion
subgroup is isomorphic to $G$ in the case of imaginary quadratic fields.\n
\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zuhair Mullath (University of Massachusetts\, Amherst)
DTSTART;VALUE=DATE-TIME:20210218T220000Z
DTEND;VALUE=DATE-TIME:20210218T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/27
DESCRIPTION:Title: Unobstructed Galois deformation problems associated to GSp(4)\nby Zu
hair Mullath (University of Massachusetts\, Amherst) as part of UCSD numbe
r theory seminar\n\nLecture held in normally APM 7321\, currently online.\
n\nAbstract\nTo a cuspidal automorphic representation of GSp(4) over $\\ma
thbb Q$\, one can associate a compatible system of Galois representations
$\\{\\rho_p\\}_{p \\\; \\mathrm{prime}}$. For $p$ sufficiently large\, the
deformation theory of the mod-$p$ reduction $\\overline \\rho_p$ is expec
ted to be unobstructed -- meaning the universal deformation ring is a powe
r series ring. The global obstructions to deforming $\\overline \\rho_p$ i
s controlled by certain adjoint Bloch-Kato Selmer groups\, which are expec
ted to be trivial for $p$ large enough.\n\nI will talk about some recent r
esults showing that there are no local obstructions to the deformation the
ory of $\\overline \\rho_p$ for almost all $p$. This is joint work with M.
Broshi\, C. Sorensen\, and T. Weston.\n\nPre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Trudgian (UNSW Canberra at ADFA)
DTSTART;VALUE=DATE-TIME:20210225T220000Z
DTEND;VALUE=DATE-TIME:20210225T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/28
DESCRIPTION:Title: Verifying the Riemann hypothesis to a new height\nby Tim Trudgian (U
NSW Canberra at ADFA) as part of UCSD number theory seminar\n\nLecture hel
d in normally APM 7321\, currently online.\n\nAbstract\nSadly\, I won’t
have time to prove the Riemann hypothesis in this talk. However\, I do hop
e to outline recent work in a record partial-verification of RH. This is j
oint work with Dave Platt\, in Bristol\, UK.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumya Sankar (The Ohio State University)
DTSTART;VALUE=DATE-TIME:20210304T220000Z
DTEND;VALUE=DATE-TIME:20210304T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/29
DESCRIPTION:Title: Counting elliptic curves with a rational $N$-isogeny\nby Soumya Sank
ar (The Ohio State University) as part of UCSD number theory seminar\n\nLe
cture held in normally APM 7321\, currently online.\n\nAbstract\nThe class
ical problem of counting elliptic curves with a rational N-isogeny can be
phrased in terms of counting rational points on certain moduli stacks of e
lliptic curves. Counting points on stacks poses various challenges\, and I
will discuss these along with a few ways to overcome them. I will also ta
lk about the theory of heights on stacks developed in recent work of Ellen
berg\, Satriano and Zureick-Brown and use it to count elliptic curves with
an $N$-isogeny for certain $N$. The talk assumes no prior knowledge of st
acks and is based on joint work with Brandon Boggess.\n\nThere will be a 3
0 minute pre-talk for graduate students and postdocs preceding the main ta
lk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Organizational meeting (UCSD)
DTSTART;VALUE=DATE-TIME:20210311T220000Z
DTEND;VALUE=DATE-TIME:20210311T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/30
DESCRIPTION:Title: Organizational meeting\nby Organizational meeting (UCSD) as part of
UCSD number theory seminar\n\nLecture held in normally APM 7321\, currentl
y online.\n\nAbstract\nOrganizational meeting to plan for next quarter. No
talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (MPIM)
DTSTART;VALUE=DATE-TIME:20210401T180000Z
DTEND;VALUE=DATE-TIME:20210401T190000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/31
DESCRIPTION:Title: Malle's conjecture for nonic Heisenberg extensions\nby Peter Koymans
(MPIM) as part of UCSD number theory seminar\n\nLecture held in normally
APM 7321\, currently online.\n\nAbstract\nIn 2002 Malle conjectured an asy
mptotic formula for the number of $G$-extensions of a number field $K$ wit
h discriminant bounded by $X$. In this talk I will discuss recent joint wo
rk with Etienne Fouvry on this conjecture. Our main result proves Malle's
conjecture in the special case of nonic Heisenberg extensions.\n\npre-talk
\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahesh Kakde (IISc\, Bangalore)
DTSTART;VALUE=DATE-TIME:20210408T170000Z
DTEND;VALUE=DATE-TIME:20210408T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/32
DESCRIPTION:Title: On the Brumer-Stark conjecture and applications to Hilbert's 12th proble
m\nby Mahesh Kakde (IISc\, Bangalore) as part of UCSD number theory se
minar\n\nLecture held in normally APM 7321\, currently online.\n\nAbstract
\nI will report on my joint work with Samit Dasgupta on the Brumer-Stark c
onjecture proving existence of the Brumer-Stark units and on a conjecture
of Dasgupta giving a p-adic analytic formula for these units. I will prese
nt a sketch of our proof of the Brumer-Stark conjecture and also mention a
pplications to Hilbert's 12th problem\, or explicit class field theory.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lance Miller (University of Arkansas)
DTSTART;VALUE=DATE-TIME:20210415T210000Z
DTEND;VALUE=DATE-TIME:20210415T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/33
DESCRIPTION:Title: Finiteness of quasi-canonical lifts of elliptic curves\nby Lance Mil
ler (University of Arkansas) as part of UCSD number theory seminar\n\nLect
ure held in normally APM 7321\, currently online.\n\nAbstract\nFix a prime
integer $p$. Set $R$ the completed valuation ring of the maximal unramifi
ed extension of $\\mathbb{Q}_p$. For $X := X_1(N)$ the modular curve with
$N$ at least 4 and coprime to $p$\, Buium-Poonen in 2009 showed that the
locus of canonical lifts enjoys finite intersection with preimages of fini
te rank subgroups of $E(R)$ when $E$ is an elliptic curve with a surjectio
n from $X$. This is done using Buium's theory of arithmetic ODEs\, in part
icular interesting homomorphisms $E(R) \\to R$ which are arithmetic analog
ues of Manin maps. \n\nIn this talk\, I will review the general idea behin
d this result and other applications of arithmetic jet spaces to Diophanti
ne geometry and discuss a recent analogous result for quasi-canonical lift
s\, i.e.\, those curves with Serre-Tate parameter a root of unity. Here th
e ODE Manin maps are insufficient\, so we introduce a PDE version of Buium
's theory to provide the needed maps. All of this is joint work with A. Bu
ium.\n\npre-talk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Owen Barrett (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210422T210000Z
DTEND;VALUE=DATE-TIME:20210422T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/34
DESCRIPTION:Title: The derived category of the abelian category of constructible sheaves\nby Owen Barrett (University of Chicago) as part of UCSD number theory s
eminar\n\nLecture held in normally APM 7321\, currently online.\n\nAbstrac
t\nNori proved in 2002 that given a complex algebraic variety $X$\, the bo
unded\nderived category of the abelian category of constructible sheaves o
n $X$ is\nequivalent to the usual triangulated category $D(X)$ of bounded\
nconstructible complexes on $X$.\nHe moreover showed that given any constr
uctible sheaf $\\mathcal F$ on\n$\\A^n$\, there is an injection $\\mathcal
F\\hookrightarrow\\mathcal G$ with\n$\\mathcal G$ constructible and $H^i(
\\A^n\,\\mathcal G)=0$ for $i>0$.\n\nIn this talk\, I'll discuss how to ex
tend Nori's theorem to the case of a\nvariety over an algebraically closed
field of positive characteristic\, with\nBetti constructible sheaves repl
aced by $\\ell$-adic sheaves.\nThis is the case $p=0$ of the general probl
em which asks whether the bounded\nderived category of $p$-perverse sheave
s is equivalent to $D(X)$\, resolved\naffirmatively for the middle pervers
ity by Beilinson.\n\npre-talk at 1:30pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Klevdal (University of Utah)
DTSTART;VALUE=DATE-TIME:20210429T210000Z
DTEND;VALUE=DATE-TIME:20210429T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/35
DESCRIPTION:Title: Integrality of G-local systems\nby Christian Klevdal (University of
Utah) as part of UCSD number theory seminar\n\nLecture held in normally AP
M 7321\, currently online.\n\nAbstract\nSimpson conjectured that for a red
uctive group $G$\, rigid $G$-local systems on a smooth projective complex
variety are integral. I will discuss a proof of integrality for cohomologi
cally rigid $G$-local systems. This generalizes and is inspired by work of
Esnault and Groechenig for $GL_n$. Surprisingly\, the main tools used in
the proof (for general $G$ and $GL_n$) are the work of L. Lafforgue on the
Langlands program for curves over function fields\, and work of Drinfeld
on companions of $\\ell$-adic sheaves. The major differences between gener
al $G$ and $GL_n$ are first to make sense of companions for $G$-local syst
ems\, and second to show that the monodromy group of a rigid G-local syste
m is semisimple. All work is joint with Stefan Patrikis.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Fox (University of Oregon)
DTSTART;VALUE=DATE-TIME:20210506T210000Z
DTEND;VALUE=DATE-TIME:20210506T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/36
DESCRIPTION:Title: Supersingular Loci of Some Unitary Shimura Varieties\nby Maria Fox (
University of Oregon) as part of UCSD number theory seminar\n\nLecture hel
d in normally APM 7321\, currently online.\n\nAbstract\nUnitary Shimura va
rieties are moduli spaces of abelian varieties with an action of a quadrat
ic imaginary field\, and extra structure. In this talk\, we'll discuss spe
cific examples of unitary Shimura varieties whose supersingular loci can b
e concretely described in terms of Deligne-Lusztig varieties. By Rapoport-
Zink uniformization\, much of the structure of these supersingular loci ca
n be understood by studying an associated moduli space of p-divisible grou
ps (a Rapoport-Zink space). We'll discuss the geometric structure of these
associated Rapoport-Zink spaces as well as some techniques for studying t
hem.\n\nThere will be a pre-talk!\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean Howe (University of Utah)
DTSTART;VALUE=DATE-TIME:20210513T210000Z
DTEND;VALUE=DATE-TIME:20210513T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/37
DESCRIPTION:Title: Bialgebraicity in local Shimura varieties\nby Sean Howe (University
of Utah) as part of UCSD number theory seminar\n\nLecture held in normally
APM 7321\, currently online.\n\nAbstract\nA classical transcendence resul
t of Schneider on the modular $j$-invariant states that\, for $\\tau \\in
\\mathbb{H}$\, both $\\tau$ and $j(\\tau)$ are in $\\overline{\\mathbb{Q}}
$ if and only if $\\tau$ is contained in an imaginary quadratic extension
of $\\mathbb{Q}$. The space $\\mathbb{H}$ has a natural interpretation as
a parameter space for $\\mathbb{Q}$-Hodge structures (or\, in this case\,
elliptic curves)\, and from this perspective the imaginary quadratic point
s are distinguished as corresponding to objects with maximal symmetry. Thi
s result has been generalized by Cohen and Shiga-Wolfart to more general m
oduli of Hodge structures (corresponding to abelian-type Shimura varieties
)\, and by Ullmo-Yafaev to higher dimensional loci of extra symmetry (spec
ial subvarieties)\, where bialgebraicity is intimately connected with the
Pila-Zannier approach to the Andre-Oort conjecture.\n\nIn this talk\, I wi
ll discuss work in progress with Christian Klevdal on an analogous bialgeb
raicity characterization of special subvarieties in Scholze's local Shimur
a varieties and more general diamond moduli of $p$-adic Hodge structures.\
n\nThere will be a pretalk!\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nahid Walji (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20210520T210000Z
DTEND;VALUE=DATE-TIME:20210520T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/38
DESCRIPTION:Title: On the conjectural decomposition of symmetric powers of automorphic repr
esentations for GL(3) and GL(4)\nby Nahid Walji (University of British
Columbia) as part of UCSD number theory seminar\n\nLecture held in normal
ly APM 7321\, currently online.\n\nAbstract\nLet $\\Pi$ be a cuspidal auto
morphic representation for GL(3) over a number field. We fix an integer $k
\\geq 2$ and we assume that the symmetric $m$th power lifts of $\\Pi$ are
automorphic for $m \\leq k$\, cuspidal for $m < k$\, and that certain ass
ociated Rankin–Selberg products are automorphic. In this setting\, we bo
und the number of cuspidal isobaric summands in the $k$th symmetric power
lift. In particular\, we show it is bounded above by 3 for $k \\geq 7$\, a
nd bounded above by 2 when $k \\geq 19$ with $k$ congruent to 1 mod 3. We
will also discuss the analogous problem for GL(4).\n\nThis will include a
pre-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evan O'Dorney (Princeton University)
DTSTART;VALUE=DATE-TIME:20210527T210000Z
DTEND;VALUE=DATE-TIME:20210527T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/39
DESCRIPTION:Title: Arithmetic statistics of $H^1(K\, T)$\nby Evan O'Dorney (Princeton U
niversity) as part of UCSD number theory seminar\n\nLecture held in normal
ly APM 7321\, currently online.\n\nAbstract\nCoclasses in a Galois cohomol
ogy group $H^1(K\, T)$ parametrize extensions of a number field with certa
in Galois group. It is natural to want to count these coclasses with gener
al local conditions and with respect to a discriminant-like invariant. In
joint work with Brandon Alberts\, I present a novel tool for studying this
: harmonic analysis on adelic cohomology\, modeled after the celebrated us
e of harmonic analysis on the adeles in Tate's thesis. This leads to a mor
e illuminating explanation of a fact previously noticed by Alberts\, namel
y that the Dirichlet series counting such coclasses is a finite sum of Eul
er products\; and we nail down the asymptotic count of coclasses in satisf
ying generality.\n\nIn the pre-talk\, I will give a rundown on the needed
background in Galois cohomology\, etale algebras\, the local Tate pairing\
, and Poitou-Tate duality.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly Isham (University of California Irvine)
DTSTART;VALUE=DATE-TIME:20210603T210000Z
DTEND;VALUE=DATE-TIME:20210603T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/40
DESCRIPTION:Title: Asymptotic growth of orders in a fixed number field via subrings in $\\m
athbb{Z}^n$\nby Kelly Isham (University of California Irvine) as part
of UCSD number theory seminar\n\nLecture held in normally APM 7321\, curre
ntly online.\n\nAbstract\nLet $K$ be a number field of degree $n$ and $\\m
athcal{O}_K$ be its ring of integers. An order in $\\mathcal{O}_K$ is a fi
nite index subring that contains the identity. A major open question in ar
ithmetic statistics asks for the asymptotic growth of orders in $K$. In th
is talk\, we will give the best known lower bound for this asymptotic grow
th. The main strategy is to relate orders in $\\mathcal{O}_K$ to subrings
in $\\mathbb{Z}^n$ via zeta functions. Along the way\, we will give lower
bounds for the asymptotic growth of subrings in $\\mathbb{Z}^n$ and for th
e number of index $p^e$ subrings in $\\mathbb{Z}^n$. We will also discuss
analytic properties of these zeta functions.\n\nThere will be a pretalk at
1:30 Pacific time.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiran Kedlaya (UCSD)
DTSTART;VALUE=DATE-TIME:20211007T210000Z
DTEND;VALUE=DATE-TIME:20211007T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/42
DESCRIPTION:Title: Orders of abelian varieties over F_2\nby Kiran Kedlaya (UCSD) as par
t of UCSD number theory seminar\n\nLecture held in APM 7321 and online.\n\
nAbstract\nWe describe several recent results on orders of abelian varieti
es over $\\mathbb{F}_2$: every positive integer occurs as the order of an
ordinary abelian variety over $\\mathbb{F}_2$ (joint with E. Howe)\; every
positive integer occurs infinitely often as the order of a simple abelian
variety over $\\mathbb{F}_2$\; the geometric decomposition of the simple
abelian varieties over $\\mathbb{F}_2$ can be described explicitly (joint
with T. D'Nelly-Warady)\; and the relative class number one problem for fu
nction fields is reduced to a finite computation (work in progress).\n\nAl
l of these results rely on the relationship between isogeny classes of abe
lian varieties over finite fields and Weil polynomials given by the work o
f Weil and Honda-Tate. With these results in hand\, most of the work is to
construct algebraic integers satisfying suitable archimedean constraints.
\n\nTalk to be given in person and streamed via Zoom.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeff Lagarias (Michigan)
DTSTART;VALUE=DATE-TIME:20211014T210000Z
DTEND;VALUE=DATE-TIME:20211014T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/43
DESCRIPTION:Title: Complex Equiangular Lines and the Stark Conjectures\nby Jeff Lagaria
s (Michigan) as part of UCSD number theory seminar\n\nLecture held in APM
7321 and online.\n\nAbstract\nThis talk is expository. It describes the hi
story of an exciting connection made by physicists between an unsolved \n
problem in combinatorial design theory- the existence of maximal sets of $
d^2$ complex equiangular lines in ${\\mathbb C}^d$-\nrephrased as a probl
em in quantum information theory\, and topics\n in algebraic number theory
involving class fields of real quadratic fields. Work of my former studen
t\nGene Kopp recently uncovered a surprising\, deep (unproved!) connectio
n with\nthe Stark conjectures. For infinitely many dimensions $d$ he pred
icts the existence of maximal equiangular sets\, \nconstructible by a spec
ific recipe starting from suitable Stark units\, in the rank one case. Num
erically computing\nspecial values at $s=0$ of suitable L-functions then p
ermits recovering the units numerically to high precision\, \nthen reconst
ructing them exactly\, then testing they satisfy suitable extra algebraic
identities to yield a construction\nof the set of equiangular lines. It h
as been carried out for $d=5\, 11\, 17$ and $23$.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Grubb (UCSD)
DTSTART;VALUE=DATE-TIME:20211021T210000Z
DTEND;VALUE=DATE-TIME:20211021T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/44
DESCRIPTION:Title: A cut-by-curves criterion for overconvergence of $F$-isocrystals\nby
Thomas Grubb (UCSD) as part of UCSD number theory seminar\n\nLecture held
in APM 7321 and online.\n\nAbstract\nLet $X$ be a smooth\, geometrically
irreducible scheme over a finite field of characteristic $p > 0$. With res
pect to rigid cohomology\, $p$-adic coefficient objects on $X$ come in two
types: convergent $F$-isocrystals and the subcategory of overconvergent $
F$-isocrystals. Overconvergent isocrystals are related to $\\ell$-adic eta
le objects ($\\ell\\neq p$) via companions theory\, and as such it is desi
rable to understand when an isocrystal is overconvergent. We show (under a
geometric tameness hypothesis) that a convergent $F$-isocrystal $E$ is ov
erconvergent if and only if its restriction to all smooth curves on $X$ is
. The technique reduces to an algebraic setting where we use skeleton shea
ves and crystalline companions to compare $E$ to an isocrystal which is pa
tently overconvergent. Joint with Kiran Kedlaya and James Upton.\n\npre-ta
lk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rahul Dalal (Johns Hopkins)
DTSTART;VALUE=DATE-TIME:20211028T210000Z
DTEND;VALUE=DATE-TIME:20211028T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/45
DESCRIPTION:Title: Counting level-1\, quaternionic automorphic representations on $G_2$
\nby Rahul Dalal (Johns Hopkins) as part of UCSD number theory seminar\n\n
Lecture held in APM 7321 and online.\n\nAbstract\nQuaternionic automorphic
representations are one attempt to generalize to other groups the special
place holomorphic modular forms have among automorphic representations of
$GL_2$. Like holomorphic modular forms\, they are defined by having their
real component be one of a particularly nice class (in this case\, called
quaternionic discrete series). We count quaternionic automorphic represen
tations on the exceptional group $G_2$ by developing a $G_2$ version of th
e classical Eichler-Selberg trace formula for holomorphic modular forms. \
n\nThere are two main technical difficulties. First\, quaternionic discret
e series come in L-packets with non-quaternionic members and standard inva
riant trace formula techniques cannot easily distinguish between discrete
series with real component in the same L-packet. Using the more modern sta
ble trace formula resolves this issue. Second\, quaternionic discrete seri
es do not satisfy a technical condition of being "regular"\, so the trace
formula can a priori pick up unwanted contributions from automorphic repre
sentations with non-tempered components at infinity. Applying some computa
tions of Mundy\, this miraculously does not happen for our specific case o
f quaternionic representations on $G_2$. \n\nFinally\, we are only studyin
g level-1 forms\, so we can apply some tricks of Chenevier and Taïbi to r
educe the problem to counting representations on the compact form of $G_2$
and certain pairs of modular forms. This avoids involved computations on
the geometric side of the trace formula.\n\n30 min pre-talk before\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linus Hamann (Princeton)
DTSTART;VALUE=DATE-TIME:20211104T210000Z
DTEND;VALUE=DATE-TIME:20211104T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/46
DESCRIPTION:Title: Compatibility of the Fargues-Scholze and Gan-Takeda local Langlands\
nby Linus Hamann (Princeton) as part of UCSD number theory seminar\n\nLect
ure held in APM 6402 and online.\n\nAbstract\nGiven a prime $p$\, a finite
extension $L/\\mathbb{Q}_{p}$\, a connected $p$-adic reductive group $G/L
$\, and a smooth irreducible representation $\\pi$ of $G(L)$\, Fargues-Sch
olze recently attached a semisimple Weil parameter to such $\\pi$\, giving
a general candidate for the local Langlands correspondence. It is natural
to ask whether this construction is compatible with known instances of th
e correspondence after semisimplification. For $G = GL_{n}$ and its inner
forms\, Fargues-Scholze and Hansen-Kaletha-Weinstein show that the corres
pondence is compatible with the correspondence of Harris-Taylor/Henniart.
We verify a similar compatibility for $G = GSp_{4}$ and its unique non-spl
it inner form $G = GU_{2}(D)$\, where $D$ is the quaternion division algeb
ra over $L$\, assuming that $L/\\mathbb{Q}_{p}$ is unramified and $p > 2$.
In this case\, the local Langlands correspondence has been constructed by
Gan-Takeda and Gan-Tantono. Analogous to the case of $GL_{n}$ and its inn
er forms\, this compatibility is proven by describing the Weil group actio
n on the cohomology of a local Shimura variety associated to $GSp_{4}$\, u
sing basic uniformization of abelian type Shimura varieties due to Shen\,
combined with various global results of Kret-Shin and Sorensen on Galois r
epresentations in the cohomology of global Shimura varieties associated to
inner forms of $GSp_{4}$ over a totally real field. After showing the par
ameters are the same\, we apply some ideas from the geometry of the Fargue
s-Scholze construction explored recently by Hansen\, to give a more precis
e description of the cohomology of this local Shimura variety\, verifying
a strong form of the Kottwitz conjecture in the process.\n\npre-talk at 1:
20pm.\n\nThe talk will be given via Zoom\, but we will meet in the lecture
hall as usual.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriel Dorfsman-Hopkins (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20211118T220000Z
DTEND;VALUE=DATE-TIME:20211118T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/47
DESCRIPTION:Title: Untilting Line Bundles on Perfectoid Spaces\nby Gabriel Dorfsman-Hop
kins (UC Berkeley) as part of UCSD number theory seminar\n\nLecture held i
n APM 7321 and online.\n\nAbstract\nLet $X$ be a perfectoid space with til
t $X^\\flat$. We build a natural map $\\theta:\\Pic X^\\flat\\to\\lim\\Pi
c X$ where the (inverse) limit is taken over the $p$-power map\, and show
that $\\theta$ is an isomorphism if $R = \\Gamma(X\,\\sO_X)$ is a perfecto
id ring. As a consequence we obtain a characterization of when the Picard
groups of $X$ and $X^\\flat$ agree in terms of the $p$-divisibility of $\
\Pic X$. The main technical ingredient is the vanishing of higher derived
limits of the unit group $R^*$\, whence the main result follows from the
Grothendieck spectral sequence.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Upton (UC San Diego)
DTSTART;VALUE=DATE-TIME:20211202T220000Z
DTEND;VALUE=DATE-TIME:20211202T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/48
DESCRIPTION:Title: Newton Polygons of Abelian $L$-Functions on Curves\nby James Upton (
UC San Diego) as part of UCSD number theory seminar\n\nLecture held in APM
7321 and online.\n\nAbstract\nLet $X$ be a smooth\, affine\, geometricall
y connected curve over a finite field of characteristic $p > 2$. Let $\\rh
o:\\pi_1(X) \\to \\mathbb{C}^\\times$ be a character of finite order $p^n$
. If $\\rho\\neq 1$\, then the Artin $L$-function $L(\\rho\,s)$ is a polyn
omial\, and a theorem of Kramer-Miller states that its $p$-adic Newton pol
ygon $\\mathrm{NP}(\\rho)$ is bounded below by a certain Hodge polygon $\\
mathrm{HP}(\\rho)$ which is defined in terms of local monodromy invariants
. In this talk we discuss the interaction between the polygons $\\mathrm{N
P}(\\rho)$ and $\\mathrm{HP}(\\rho)$. Our main result states that if $X$ i
s ordinary\, then $\\mathrm{NP}(\\rho)$ and $\\mathrm{HP}(\\rho)$ share a
vertex if and only if there is a corresponding vertex shared by certain "l
ocal" Newton and Hodge polygons associated to each ramified point of $\\rh
o$. As an application\, we give a local criterion that is necessary and su
fficient for $\\mathrm{NP}(\\rho)$ and $\\mathrm{HP}(\\rho)$ to coincide.
This is joint work with Joe Kramer-Miller.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Organizational meeting (UCSD)
DTSTART;VALUE=DATE-TIME:20210923T210000Z
DTEND;VALUE=DATE-TIME:20210923T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/49
DESCRIPTION:Title: Organizational meeting (Zoom only)\nby Organizational meeting (UCSD)
as part of UCSD number theory seminar\n\nLecture held in APM 7321 and onl
ine.\n\nAbstract\nThis meeting will take place exclusively over Zoom.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Organizational meeting
DTSTART;VALUE=DATE-TIME:20220106T220000Z
DTEND;VALUE=DATE-TIME:20220106T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/50
DESCRIPTION:Title: Organizational meeting (Zoom only)\nby Organizational meeting as par
t of UCSD number theory seminar\n\nLecture held in APM 6402 and online.\nA
bstract: TBA\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tong Liu (Purdue)
DTSTART;VALUE=DATE-TIME:20220113T220000Z
DTEND;VALUE=DATE-TIME:20220113T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/51
DESCRIPTION:Title: Prismatic F-crystal and lattice in crystalline representation\nby To
ng Liu (Purdue) as part of UCSD number theory seminar\n\n\nAbstract\nIn th
is talk\, I will explain a theorem of Bhatt-Scholze: the equivalence betwe
en prismatic $F$-crystal and $\\mathbb Z_p$-lattices inside crystalline re
presentation\, and how to extend this theorem to allow more general types
of base ring like Tate algebra ${\\mathbb Z}_p \\langle t^{\\pm 1}\\rangle
$. This is a joint work with Heng Du\, Yong-Suk Moon and Koji Shimizu. \
n\nThis is a talk in integral $p$-adic Hodge theory. So in the pre-talk\,
I will explain the motivations and base ideas in integral $p$-adic Hodge
theory.\n\nonline only\; pre-talk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudius Heyer (Münster)
DTSTART;VALUE=DATE-TIME:20220120T220000Z
DTEND;VALUE=DATE-TIME:20220120T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/52
DESCRIPTION:Title: The left adjoint of derived parabolic induction\nby Claudius Heyer (
Münster) as part of UCSD number theory seminar\n\n\nAbstract\nRecent adva
nces in the theory of smooth mod $p$ representations of a $p$-adic\nreduct
ive group $G$ involve more and more derived methods. It becomes\nincreasi
ngly clear that the proper framework to study smooth mod $p$\nrepresentati
ons is the derived category $D(G)$.\n\nI will talk about smooth mod $p$ re
presentations and highlight their\nshortcomings compared to\, say\, smooth
complex representations of $G$. After\nexplaining how the situation impr
oves in the derived category\, I will spend the\nremaining time on the lef
t adjoint of the derived parabolic induction functor.\n\nThere will be a p
re-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petar Bakic (Utah)
DTSTART;VALUE=DATE-TIME:20220127T220000Z
DTEND;VALUE=DATE-TIME:20220127T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/53
DESCRIPTION:Title: Howe duality for exceptional theta correspondences\nby Petar Bakic (
Utah) as part of UCSD number theory seminar\n\nLecture held in online.\n\n
Abstract\nThe theory of local theta correspondence is built up from two ma
in ingredients: a reductive dual pair inside a symplectic group\, and a We
il representation of its metaplectic cover. Exceptional correspondences ar
ise similarly: dual pairs inside exceptional groups can be constructed usi
ng so-called Freudenthal Jordan algebras\, while the minimal representatio
n provides a suitable replacement for the Weil representation. The talk wi
ll begin by recalling these constructions. Focusing on a particular dual p
air\, we will explain how one obtains Howe duality for the correspondence
in question. Finally\, we will discuss applications of these results. The
new work in this talk is joint with Gordan Savin.\n\npre-talk at 1:30pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Smith (Stanford)
DTSTART;VALUE=DATE-TIME:20220203T220000Z
DTEND;VALUE=DATE-TIME:20220203T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/54
DESCRIPTION:Title: $2^k$-Selmer groups and Goldfeld's conjecture\nby Alex Smith (Stanfo
rd) as part of UCSD number theory seminar\n\nLecture held in APM 6402 and
online.\n\nAbstract\nTake $E$ to be an elliptic curve over a number field
whose four torsion obeys certain technical conditions. In this talk\, we w
ill outline a proof that 100% of the quadratic twists of $E$ have rank at
most one. To do this\, we will find the distribution of $2^k$-Selmer ranks
in this family for every positive $k$. We will also show how are techniqu
es may be applied to find the distribution of $2^k$-class groups of quadra
tic fields.\n\nThe pre-talk will focus on the definition of Selmer groups.
We will also give some context for the study of the arithmetic statistics
of these groups.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabrielle De Micheli (UCSD)
DTSTART;VALUE=DATE-TIME:20220210T220000Z
DTEND;VALUE=DATE-TIME:20220210T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/55
DESCRIPTION:Title: Lattice Enumeration for Tower NFS: a 521-bit Discrete Logarithm Computat
ion\nby Gabrielle De Micheli (UCSD) as part of UCSD number theory semi
nar\n\nLecture held in APM 6402 and online.\n\nAbstract\nThe Tower variant
of the Number Field Sieve (TNFS) is known to be asymptotically the most e
fficient algorithm to solve the discrete logarithm problem in finite field
s of medium characteristics\, when the extension degree is composite. A ma
jor obstacle to an efficient implementation of TNFS is the collection of a
lgebraic relations\, as it happens in dimensions greater than 2. This requ
ires the construction of new sieving algorithms which remain efficient as
the dimension grows. In this talk\, I will present how we overcome this d
ifficulty by considering a lattice enumeration algorithm which we adapt to
this specific context. We also consider a new sieving area\, a high-dimen
sional sphere\, whereas previous sieving algorithms for the classical NFS
considered an orthotope. Our new sieving technique leads to a much smaller
running time\, despite the larger dimension of the search space\, and eve
n when considering a larger target\, as demonstrated by a record computati
on we performed in a 521-bit finite field GF(p^6). The target finite field
is of the same form as finite fields used in recent zero-knowledge proofs
in some blockchains. This is the first reported implementation of TNFS.\n
\nIn the pre-talk\, I will briefly present the core ideas of the quadratic
sieve algorithm and its evolution to the Number Field Sieve algorithm.\n\
npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aaron Pollack (UCSD)
DTSTART;VALUE=DATE-TIME:20220217T220000Z
DTEND;VALUE=DATE-TIME:20220217T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/56
DESCRIPTION:Title: A Cohen-Zagier modular form on G_2\nby Aaron Pollack (UCSD) as part
of UCSD number theory seminar\n\nLecture held in APM 6402 and online.\n\nA
bstract\nI will report on joint work with Spencer Leslie where we define a
n analogue of the Cohen-Zagier Eisenstein series to the exceptional group
$G_2$. Recall that the Cohen-Zagier Eisenstein series is a weight $3/2$ mo
dular form whose Fourier coefficients see the class numbers of imaginary q
uadratic fields. We define a particular modular form of weight $1/2$ on $G
_2$\, and prove that its Fourier coefficients see (certain torsors for) th
e 2-torsion in the narrow class groups of totally real cubic fields. In pa
rticular: 1) we define a notion of modular forms of half-integral weight o
n certain exceptional groups\, 2) we prove that these modular forms have a
nice theory of Fourier coefficients\, and 3) we partially compute the Fou
rier coefficients of a particular nice example on G_2.\n\npre-talk at 1:20
pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paulina Fust (Duisburg-Essen)
DTSTART;VALUE=DATE-TIME:20220224T220000Z
DTEND;VALUE=DATE-TIME:20220224T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/57
DESCRIPTION:Title: Continuous group cohomology and Ext-groups\nby Paulina Fust (Duisbur
g-Essen) as part of UCSD number theory seminar\n\nLecture held in APM 6402
and online.\n\nAbstract\nWe prove that the continuous cohomology groups o
f a $p$-adic reductive group with coefficients in an admissible unitary $\
\mathbb{Q}_p$-Banach space representation $\\Pi$ are finite-dimensional an
d compare them to certain Ext-groups. As an application of this result\, w
e show that the continuous cohomology of $SL_2(\\mathbb{Q}_p) $ with value
s in non-ordinary irreducible $\\mathbb{Q}_p$-Banach space representations
of $GL_2(\\mathbb{Q}_p)$ vanishes.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annie Carter (UCSD)
DTSTART;VALUE=DATE-TIME:20220303T170000Z
DTEND;VALUE=DATE-TIME:20220303T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/58
DESCRIPTION:Title: Two-variable polynomials with dynamical Mahler measure zero\nby Anni
e Carter (UCSD) as part of UCSD number theory seminar\n\n\nAbstract\nIntro
duced by Lehmer in 1933\, the classical Mahler measure of a complex ration
al function $P$ in one or more variables is given by integrating $\\log|P(
x_1\, \\ldots\, x_n)|$ over the unit torus. Lehmer asked whether the Mahle
r measures of integer polynomials\, when nonzero\, must be bounded away fr
om zero\, a question that remains open to this day. In this talk we genera
lize Mahler measure by associating it with a discrete dynamical system $f:
\\mathbb{C} \\to \\mathbb{C}$\, replacing the unit torus by the $n$-fold
Cartesian product of the Julia set of $f$ and integrating with respect to
the equilibrium measure on the Julia set. We then characterize those two-v
ariable integer polynomials with dynamical Mahler measure zero\, condition
al on a dynamical version of Lehmer's conjecture.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Urbanik (Toronto)
DTSTART;VALUE=DATE-TIME:20220310T220000Z
DTEND;VALUE=DATE-TIME:20220310T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/59
DESCRIPTION:Title: Effective Methods for Shafarevich Problems\nby David Urbanik (Toront
o) as part of UCSD number theory seminar\n\nLecture held in APM 7321.\n\nA
bstract\nGiven a smooth projective family $f : X \\to S$ defined over\nthe
ring of integers of a number field\, the Shafarevich problem is to\ndescr
ibe those fibres of f which have everywhere good reduction. This\ncan be i
nterpreted as asking for the dimension of the Zariski closure\nof the set
of integral points of $S$\, and is ultimately a difficult\ndiophantine que
stion about which little is known as soon as the\ndimension of $S$ is at l
east 2. Recently\, Lawrence and Venkatesh gave a\ngeneral strategy for add
ressing such problems which requires as input\nlower bounds on the monodro
my of f over essentially arbitrary closed\nsubvarieties of $S$. In this ta
lk we review their ideas\, and describe\nrecent work which gives a fully e
ffective method for computing these\nlower bounds. This gives a fully effe
ctive strategy for solving\nShafarevich-type problems for essentially arbi
trary families $f$.\n\nThis week's talk is in APM 7321 rather than APM 640
2.\n\npre-talk at 1:20 pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amit Ophir (Hebrew U.)
DTSTART;VALUE=DATE-TIME:20220414T170000Z
DTEND;VALUE=DATE-TIME:20220414T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/60
DESCRIPTION:Title: Invariant norms on the p-adic Schrödinger representation\nby Amit O
phir (Hebrew U.) as part of UCSD number theory seminar\n\nLecture held in
online.\n\nAbstract\nMotivated by questions about a p-adic Fourier transfo
rm\, we study invariant norms on the p-adic Schrödinger representations o
f Heisenberg groups. These Heisenberg groups are p-adic\, and the Schrodin
ger representations are explicit irreducible smooth representations that p
lay an important role in their representation theory. \nClassically\, the
field of coefficients is taken to be the complex numbers and\, among other
things\, one studies the unitary completions of the representations (whic
h are well understood). By taking the field of coefficients to be an exten
sion of the p-adic numbers\, we can consider completions that better captu
re the p-adic topology\, but at the cost of losing the Haar measure and th
e $L^2$-norm. Nevertheless\, we establish a rigidity property for a family
of norms (parametrized by a Grassmannian) that are invariant under the ac
tion of the Heisenberg group.\nThe irreducibility of some Banach represent
ations follows as a result. The proof uses "q-arithmetics".\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Lawrence (UCLA)
DTSTART;VALUE=DATE-TIME:20220428T210000Z
DTEND;VALUE=DATE-TIME:20220428T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/61
DESCRIPTION:Title: Sparsity of Integral Points on Moduli Spaces of Varieties\nby Brian
Lawrence (UCLA) as part of UCSD number theory seminar\n\nLecture held in A
PM 6402 and online.\n\nAbstract\nInteresting moduli spaces don't have many
integral points. More precisely\, if X is a variety over a number field\
, admitting a variation of Hodge structure whose associate period map is i
njective\, then the number of S-integral points on X of height at most H g
rows more slowly than H^{\\epsilon}\, for any positive \\epsilon. This is
a sort of weak generalization of the Shafarevich conjecture\; it is a con
sequence of a point-counting theorem of Broberg\, and the largeness of the
fundamental group of X. Joint with Ellenberg and Venkatesh.\n\npre-talk
at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michelle Manes (U. Hawaii)
DTSTART;VALUE=DATE-TIME:20220519T210000Z
DTEND;VALUE=DATE-TIME:20220519T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/62
DESCRIPTION:Title: Iterating Backwards in Arithmetic Dynamics\nby Michelle Manes (U. Ha
waii) as part of UCSD number theory seminar\n\nLecture held in APM 6402 an
d online.\n\nAbstract\nIn classical real and complex dynamics\, one studie
s topological and analytic properties of orbits of points under iteration
of self-maps of $\\mathbb R$ or $\\mathbb C$ (or more generally self-maps
of a real or complex manifold). In arithmetic dynamics\, a more recent sub
ject\, one likewise studies properties of orbits of self-maps\, but with a
number theoretic flavor. Many of the motivating problems in arithmetic dy
namics come via analogy with classical problems in arithmetic geometry: ra
tional and integral points on varieties correspond to rational and integra
l points in orbits\; torsion points on abelian varieties correspond to per
iodic and preperiodic points of rational maps\; and abelian varieties with
complex multiplication correspond to post-critically finite rational maps
.\n\nThis analogy focuses on forward iteration\, but sometimes surprising
and interesting results can be found by thinking instead about pre-images
of rational points under iteration. In this talk\, we will give some backg
round and motivation for the field of arithmetic dynamics in order to desc
ribe some of these "backwards iteration" results\, including uniform bound
edness for rational pre-images and open image results for Galois represent
ations associated to dynamical systems.\n\nA pre-talk for graduate student
s will describe some of the motivating results in arithmetic geometry.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Koji Shimizu (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20220526T210000Z
DTEND;VALUE=DATE-TIME:20220526T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/63
DESCRIPTION:Title: Robba site and Robba cohomology\nby Koji Shimizu (UC Berkeley) as pa
rt of UCSD number theory seminar\n\nLecture held in APM 6402 and online.\n
\nAbstract\nWe will discuss a $p$-adic cohomology theory for rigid analyti
c varieties with overconvergent structure (dagger spaces) over a local fie
ld of characteristic $p$. After explaining the motivation\, we will define
a site (Robba site) and discuss its basic properties.\n\nThe main talk wi
ll be preceded by a pre-talk.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Hansen (MPIM)
DTSTART;VALUE=DATE-TIME:20220407T210000Z
DTEND;VALUE=DATE-TIME:20220407T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/64
DESCRIPTION:Title: Duality and the p-adic Jacquet-Langlands correspondence\nby David Ha
nsen (MPIM) as part of UCSD number theory seminar\n\nLecture held in APM 6
402 and online.\n\nAbstract\nIn joint work with Lucas Mann\, we establish
several new properties of the p-adic Jacquet-Langlands functor defined by
Scholze in terms of the cohomology of the Lubin-Tate tower. In particular\
, we prove a duality theorem\, establish bounds on Gelfand-Kirillov dimens
ion\, prove some non-vanishing results\, and show a kind of partial Künne
th formula. The key new tool is the six functor formalism with solid almos
t $\\mathcal{O}^+/p$-coefficients developed recently by Mann.\n\nPre-talk\
n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Organizational meeting
DTSTART;VALUE=DATE-TIME:20220331T210000Z
DTEND;VALUE=DATE-TIME:20220331T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/65
DESCRIPTION:Title: Organizational meeting (Zoom only)\nby Organizational meeting as par
t of UCSD number theory seminar\n\nLecture held in online.\nAbstract: TBA\
n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Kling (U. Arizona)
DTSTART;VALUE=DATE-TIME:20220421T210000Z
DTEND;VALUE=DATE-TIME:20220421T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/66
DESCRIPTION:Title: Comparison of Integral Structures on the Space of Modular Forms of Full
Level $N$\nby Anthony Kling (U. Arizona) as part of UCSD number theory
seminar\n\nLecture held in APM 6402 and online.\n\nAbstract\nLet $N\\geq3
$ and $r\\geq1$ be integers and $p\\geq2$ be a prime such that $p\\nmid N$
. One can consider two different integral structures on the space of modul
ar forms over $\\mathbb{Q}$\, one coming from arithmetic via $q$-expansion
s\, the other coming from geometry via integral models of modular curves.
Both structures are stable under the Hecke operators\; furthermore\, their
quotient is finite torsion. Our goal is to investigate the exponent of th
e annihilator of the quotient. We will apply results due to Brian Conrad t
o the situation of modular forms of even weight and level $\\Gamma(Np^{r})
$ over $\\mathbb{Q}_{p}(\\zeta_{Np^{r}})$ to obtain an upper bound for the
exponent. We also use Klein forms to construct explicit modular forms of
level $p^{r}$ whenever $p^{r}>3$\, allowing us to compute a lower bound wh
ich agrees with the upper bound. Hence we are able to compute the exponent
precisely.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Masahiro Nakahara (U. Washington)
DTSTART;VALUE=DATE-TIME:20220505T210000Z
DTEND;VALUE=DATE-TIME:20220505T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/67
DESCRIPTION:Title: Uniform potential density for rational points on algebraic groups and el
liptic K3 surfaces\nby Masahiro Nakahara (U. Washington) as part of UC
SD number theory seminar\n\nLecture held in APM 6402 and online.\n\nAbstra
ct\nA variety satisfies potential density if it contains a dense subset of
rational points after extending its ground field by a finite degree. A co
llection of varieties satisfies uniform potential density if that degree c
an be uniformly bounded. I will discuss this property for connected algebr
aic groups of a fixed dimension and elliptic K3 surfaces. This is joint wo
rk with Kuan-Wen Lai.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gyujin Oh (Princeton)
DTSTART;VALUE=DATE-TIME:20220512T210000Z
DTEND;VALUE=DATE-TIME:20220512T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/68
DESCRIPTION:Title: A cohomological approach to harmonic Maass forms\nby Gyujin Oh (Prin
ceton) as part of UCSD number theory seminar\n\nLecture held in APM 6402 a
nd online.\n\nAbstract\nWe interpret a harmonic Maass form as a variant of
a local cohomology class of the modular curve. This is not only amenable
to algebraic interpretation\, but also nicely generalized to other Shimura
varieties\, avoiding the barrier of Koecher's principle\, which could be
useful for developing a generalization of Borcherds lifts. In this talk\,
we will exhibit how the theory looks like in the case of Hilbert modular v
arities.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Florea (UC Irvine)
DTSTART;VALUE=DATE-TIME:20220602T210000Z
DTEND;VALUE=DATE-TIME:20220602T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/69
DESCRIPTION:Title: Negative moments of the Riemann zeta function\nby Alexandra Florea (
UC Irvine) as part of UCSD number theory seminar\n\nLecture held in APM 64
02 and online.\n\nAbstract\nI will talk about recent work towards a conjec
ture of Gonek regarding negative shifted moments of the Riemann zeta funct
ion. I will explain how to obtain asymptotic formulas when the shift in th
e Riemann zeta function is big enough\, and how we can obtain non-trivial
upper bounds for smaller shifts. Joint work with H. Bui.\n\npre-talk at 1:
20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Organizational meeting (UCSD)
DTSTART;VALUE=DATE-TIME:20220929T210000Z
DTEND;VALUE=DATE-TIME:20220929T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/70
DESCRIPTION:Title: Organizational meeting (APM 7321)\nby Organizational meeting (UCSD)
as part of UCSD number theory seminar\n\nLecture held in APM 6402 and onli
ne.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Klevdal (UCSD)
DTSTART;VALUE=DATE-TIME:20221006T210000Z
DTEND;VALUE=DATE-TIME:20221006T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/71
DESCRIPTION:Title: Strong independence of $\\ell$ for Shimura varieties\nby Christian K
levdal (UCSD) as part of UCSD number theory seminar\n\nLecture held in APM
6402 and online.\n\nAbstract\n(Joint with Stefan Patrikis.) In this talk\
, we discuss a strong form of independence of $\\ell$ for canonical $\\ell
$-adic local systems on Shimura varieties\, and sketch a proof of this for
Shimura varieties arising from adjoint groups whose simple factors have r
eal rank $\\geq 2$. Notably\, this includes all adjoint Shimura varieties
which are not of abelian type. The key tools used are the existence of com
panions for $\\ell$-adic local systems and the superrigidity theorem of Ma
rgulis for lattices in Lie groups of real rank $\\geq 2$. \n\nThe indepen
dence of $\\ell$ is motivated by a conjectural description of Shimura vari
eties as moduli spaces of motives. For certain Shimura varieties that aris
e as a moduli space of abelian varieties\, the strong independence of $\\e
ll$ is proven (at the level of Galois representations) by recent work of K
isin and Zhou\, refining the independence of $\\ell$ on the Tate module gi
ven by Deligne's work on the Weil conjectures.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shishir Agrawal (UCSD)
DTSTART;VALUE=DATE-TIME:20221013T210000Z
DTEND;VALUE=DATE-TIME:20221013T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/72
DESCRIPTION:Title: From category $\\mathcal{O}^\\infty$ to locally analytic representations
\nby Shishir Agrawal (UCSD) as part of UCSD number theory seminar\n\nL
ecture held in APM 6402 and online.\n\nAbstract\nLet $G$ be a $p$-adic red
uctive group with $\\mathfrak{g} = \\mathrm{Lie}(G)$. I will summarize wor
k with Matthias Strauch in which we construct an exact functor from catego
ry $\\mathcal{O}^\\infty$\, the extension closure of the Bernstein-Gelfand
-Gelfand category $\\mathcal{O}$ inside the category of $U(\\mathfrak{g})$
-modules\, into the category of admissible locally analytic representation
s of $G$. This expands on an earlier construction by Sascha Orlik and Matt
hias Strauch. A key role in our new construction is played by $p$-adic log
arithms on tori\, and representations in the image of this functor are rel
ated to some that are known to arise in the context of the $p$-adic Langla
nds program.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jon Aycock (UCSD)
DTSTART;VALUE=DATE-TIME:20221020T210000Z
DTEND;VALUE=DATE-TIME:20221020T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/73
DESCRIPTION:Title: Differential operators for overconvergent Hilbert modular forms\nby
Jon Aycock (UCSD) as part of UCSD number theory seminar\n\nLecture held in
APM 6402 and online.\n\nAbstract\nIn 1978\, Katz gave a construction of t
he $p$-adic $L$-function of a CM field by using a $p$-adic analog of the M
aass--Shimura operators acting on $p$-adic Hilbert modular forms. However\
, this $p$-adic Maass--Shimura operator is only defined over the ordinary
locus\, which restricted Katz's choice of $p$ to one that splits in the CM
field. In 2021\, Andreatta and Iovita extended Katz's construction to all
$p$ for quadratic imaginary fields using overconvergent differential oper
ators constructed by Harron--Xiao and Urban\, which act on elliptic modula
r forms. Here we give a construction of such overconvergent differential o
perators which act on Hilbert modular forms.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rusiru Gambheera Arachchige (UCSD)
DTSTART;VALUE=DATE-TIME:20221027T210000Z
DTEND;VALUE=DATE-TIME:20221027T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/74
DESCRIPTION:Title: An unconditional equivariant main conjecture in Iwasawa theory\nby R
usiru Gambheera Arachchige (UCSD) as part of UCSD number theory seminar\n\
nLecture held in APM 6402 and online.\n\nAbstract\nIn 2015 Greither and Po
pescu constructed a new class of Iwasawa modules\, which are the number fi
eld analogues of $p-$adic realizations of Picard 1- motives constructed by
Deligne. They proved an equivariant main conjecture by computing the Fitt
ing ideal of these new modules over the appropriate profinite group ring.
This is an integral\, equivariant refinement of Wiles' classical main conj
ecture. As a consequence they proved a refinement of the Brumer-Stark conj
ecture away from 2. All of the above was proved under the assumption that
the relevant prime $p$ is odd and that the appropriate classical Iwasawa $
\\mu$–invariants vanish. Recently\, Dasgupta and Kakde proved the Brumer
-Stark conjecture\, away from 2\, unconditionally\, using a generalization
of Ribet's method. We use the Dasgupta-Kakde results to prove an uncondit
ional equivariant main conjecture\, which is a generalization of that of G
reither and Popescu. As applications of our main theorem we prove a genera
lization of a certain case of the main result of Dasgupta-Kakde and we com
pute the Fitting ideal of a certain naturally arising Iwasawa module. This
is joint work with Cristian Popescu.\n\npre-talk at 1:20\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Finn McGlade (UCSD)
DTSTART;VALUE=DATE-TIME:20221103T210000Z
DTEND;VALUE=DATE-TIME:20221103T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/75
DESCRIPTION:Title: Fourier coefficients on quaternionic U(2\,n)\nby Finn McGlade (UCSD)
as part of UCSD number theory seminar\n\nLecture held in APM 6402 and onl
ine.\n\nAbstract\nLet $E/\\mathbb{Q}$ be an imaginary quadratic extension
and suppose $G$ is the unitary group attached to hermitian space over $E$
of signature $(2\,n)$. The symmetric domain $X$ attached to $G$ is a quate
rnionic Kahler manifold. In the late nineties N. Wallach studied harmonic
analysis on $X$ in the context of this quaternionic structure. He establis
hed a multiplicity one theorem for spaces of generalized Whittaker periods
appearing in the cohomology of certain $G$-bundles on $X$. \n\nWe prove a
n analogous multiplicity one statement for some degenerate generalized Whi
ttaker periods and give explicit formulas for these periods in terms of mo
dified K-Bessel functions. Our results give a refinement of the Fourier ex
pansion of certain automorphic forms on $G$ which are quaternionic discret
e series at infinity. As an application\, we study the Fourier expansion o
f cusp forms on $G$ which arise as theta lifts of holomorphic modular form
s on quasi-split $\\mathrm{U}(1\,1)$. We show that these cusp forms can be
normalized so that all their Fourier coefficients are algebraic numbers.
(joint with Anton Hilado and Pan Yan)\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kalyani Kansal (Johns Hopkins)
DTSTART;VALUE=DATE-TIME:20221110T220000Z
DTEND;VALUE=DATE-TIME:20221110T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/76
DESCRIPTION:Title: Intersections of components of Emerton-Gee stack for $\\mathrm{GL}_2$\nby Kalyani Kansal (Johns Hopkins) as part of UCSD number theory seminar
\n\nLecture held in APM 6402 and online.\n\nAbstract\nThe Emerton-Gee stac
k for $\\mathrm{GL}_2$ is a stack of $(\\varphi\, \\Gamma)$-modules whose
reduced part $\\mathcal{X}_{2\, \\mathrm{red}}$ can be viewed as a moduli
stack of mod $p$ representations of a $p$-adic Galois group. We compute cr
iteria for codimension one intersections of the irreducible components of
$\\mathcal{X}_{2\, \\mathrm{red}}$\, and interpret them in sheaf-theoretic
terms. We also give a cohomological criterion for the number of top-dimen
sional components in a codimension one intersection.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Romyar Sharifi (UCLA)
DTSTART;VALUE=DATE-TIME:20221117T220000Z
DTEND;VALUE=DATE-TIME:20221117T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/77
DESCRIPTION:Title: Cohomology of intermediate quotients\nby Romyar Sharifi (UCLA) as pa
rt of UCSD number theory seminar\n\nLecture held in APM 6402 and online.\n
\nAbstract\nWe will discuss Galois cohomology groups of “intermediate”
quotients of an induced module\, which sit between Iwasawa cohomology up
a tower and cohomology over the ground field. Special elements in Iwasawa
cohomology that arise from Euler systems become divisible by a certain Eul
er factor upon norming down to the ground field. In certain instances\, th
ere are reasons to wonder whether this divisibility can also hold for the
image in intermediate cohomology. Using “intermediate” Coleman maps\,
we shall see that the situation locally at $p$ is as nice as one could ima
gine.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Keyes (Emory)
DTSTART;VALUE=DATE-TIME:20221201T220000Z
DTEND;VALUE=DATE-TIME:20221201T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/78
DESCRIPTION:Title: Local solubility in families of superelliptic curves\nby Christopher
Keyes (Emory) as part of UCSD number theory seminar\n\nLecture held in AP
M 6402 and online.\n\nAbstract\nIf we choose at random an integral binary
form $f(x\, z)$ of fixed degree $d$\, what is the probability that the sup
erelliptic curve with equation $C \\colon: y^m = f(x\, z)$ has a $p$-adic
point\, or better\, points everywhere locally? In joint work with Lea Ben
eish\, we show that the proportion of forms $f(x\, z)$ for which $C$ is ev
erywhere locally soluble is positive\, given by a product of local densiti
es. By studying these local densities\, we produce bounds which are suitab
le enough to pass to the large $d$ limit. In the specific case of curves o
f the form $y^3 = f(x\, z)$ for a binary form of degree 6\, we determine t
he probability of everywhere local solubility to be 96.94%\, with the exac
t value given by an explicit infinite product of rational function express
ions.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Marshall (Wisconsin)
DTSTART;VALUE=DATE-TIME:20230209T220000Z
DTEND;VALUE=DATE-TIME:20230209T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/79
DESCRIPTION:Title: Large values of eigenfunctions on hyperbolic manifolds\nby Simon Mar
shall (Wisconsin) as part of UCSD number theory seminar\n\nLecture held in
APM 6402 and online.\n\nAbstract\nIt is a folklore conjecture that the su
p norm of a Laplace eigenfunction on a compact hyperbolic surface grows mo
re slowly than any positive power of the eigenvalue. In dimensions three
and higher\, this was shown to be false by Iwaniec-Sarnak and Donnelly. I
will present joint work with Farrell Brumley that strengthens these resul
ts\, and extends them to locally symmetric spaces associated to $\\mathrm{
SO}(p\,q)$.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga (Cambridge)
DTSTART;VALUE=DATE-TIME:20230309T220000Z
DTEND;VALUE=DATE-TIME:20230309T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/80
DESCRIPTION:Title: Arithmetic statistics via graded Lie algebras\nby Jef Laga (Cambridg
e) as part of UCSD number theory seminar\n\nLecture held in APM 6402 and o
nline.\n\nAbstract\nI will explain how various results in arithmetic stati
stics by Bhargava\, Gross\, Shankar and others on 2-Selmer groups of Jacob
ians of (hyper)elliptic curves can be organised and reproved using the the
ory of graded Lie algebras\, following earlier work of Thorne. This gives
a uniform proof of these results and yields new theorems for certain famil
ies of non-hyperelliptic curves. I will also mention some applications to
rational points on certain families of curves.\n\nThe talk will involve a
mixture of representation theory\, number theory and algebraic geometry an
d I will assume no familiarity with arithmetic statistics.\n\npre-talk at
1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Longke Tang (Princeton)
DTSTART;VALUE=DATE-TIME:20230119T220000Z
DTEND;VALUE=DATE-TIME:20230119T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/81
DESCRIPTION:Title: Prismatic Poincaré duality\nby Longke Tang (Princeton) as part of U
CSD number theory seminar\n\nLecture held in APM 6402 and online.\n\nAbstr
act\nPrismatic cohomology is a new p-adic cohomology theory introduced by
Bhatt and Scholze that specializes to various well-known cohomology theori
es such as étale\, de Rham and crystalline. I will roughly recall the pro
perties of this cohomology and explain how to prove its Poincaré duality.
\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nolan Wallach (UC San Diego)
DTSTART;VALUE=DATE-TIME:20230126T220000Z
DTEND;VALUE=DATE-TIME:20230126T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/82
DESCRIPTION:Title: The Whittaker Inversion Theorem and some applications\nby Nolan Wall
ach (UC San Diego) as part of UCSD number theory seminar\n\nLecture held i
n APM 6402 and online.\n\nAbstract\nThe Whittaker Plancherel theorem appea
red as Chapter 15 in my two volume book\, Real Reductive Groups. It was me
ant to be an application of Harish-Chandra’s Plancherel Theorem. As it
turns out\, there are serious gaps in the proof given in the books. At the
same time as I was doing my research on the subject\, Harish-Chandra was
also working on it. His approach was very different from mine and appears
as part of Volume 5 of his collected works\; which consists of three piece
s of research by Harish-Chandra that were incomplete at his death and orga
nized and edited by Gangolli and Varadarajan. Unfortunately\, it also doe
s not contain a proof of the theorem. There was a complication in the proo
f of this result that caused substantial difficulties which had to do with
the image of the analog of Harish-Chandra’s method of descent. In this
lecture I will explain how one can complete the proof using a recent resul
t of Raphael Beuzzart-Plessis. I will also give an application of the resu
lt to the Fourier transforms of automorphic functions at cusps.\n\n(This s
eminar will be given remotely\, but there will still be a live audience in
the lecture room.)\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xu Gao (UC Santa Cruz)
DTSTART;VALUE=DATE-TIME:20230316T210000Z
DTEND;VALUE=DATE-TIME:20230316T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/83
DESCRIPTION:Title: $p$-adic representations and simplicial distance in Bruhat-Tits building
s\nby Xu Gao (UC Santa Cruz) as part of UCSD number theory seminar\n\n
Lecture held in APM 6402 and online.\n\nAbstract\n$p$-adic representations
are important objects in number theory\, and stable lattices serve as a c
onnection between the study of ordinary and modular representations. These
stable lattices can be understood as stable vertices in Bruhat-Tits build
ings. From this viewpoint\, the study of fixed point sets in these buildin
gs can aid research on $p$-adic representations. The simplicial distance h
olds an important role as it connects the combinatorics of lattices and th
e geometry of root systems. In particularly\, the fixed-point sets of Moy-
Prasad subgroups are precisely the simplicial balls. In this talk\, I'll e
xplain those findings and compute their simplicial volume under certain co
nditions.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sameera Vemulapalli (Princeton)
DTSTART;VALUE=DATE-TIME:20230223T220000Z
DTEND;VALUE=DATE-TIME:20230223T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/84
DESCRIPTION:Title: Counting low degree number fields with almost prescribed successive mini
ma\nby Sameera Vemulapalli (Princeton) as part of UCSD number theory s
eminar\n\nLecture held in APM 6402 and online.\n\nAbstract\nThe successive
minima of an order in a degree n number field are n real numbers encoding
information about the Euclidean structure of the order. How many orders i
n degree n number fields are there with almost prescribed successive minim
a\, fixed Galois group\, and bounded discriminant? In this talk\, I will a
ddress this question for n = 3\, 4\, 5. The answers\, appropriately interp
reted\, turn out to be piecewise linear functions on certain convex bodies
. If time permits\, I will also discuss function field analogues of this p
roblem.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yujie Xu (MIT)
DTSTART;VALUE=DATE-TIME:20230302T220000Z
DTEND;VALUE=DATE-TIME:20230302T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/85
DESCRIPTION:Title: Hecke algebras for p-adic groups and the explicit Local Langlands Corres
pondence for G_2\nby Yujie Xu (MIT) as part of UCSD number theory semi
nar\n\nLecture held in APM 6402 and online.\n\nAbstract\nI will talk about
my recent joint work with Aubert where we prove the Local Langlands Conje
cture for G_2 (explicitly). This uses our earlier results on Hecke algebra
s attached to Bernstein components of reductive p-adic groups\, as well as
an expected property on cuspidal support\, along with a list of character
izing properties. In particular\, we obtain "mixed" L-packets containing F
-singular supercuspidals and non-supercuspidals.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arghya Sadhukhan (Maryland)
DTSTART;VALUE=DATE-TIME:20230112T220000Z
DTEND;VALUE=DATE-TIME:20230112T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/86
DESCRIPTION:Title: Understanding the dimension of some (union of) affine Deligne-Lusztig va
rieties via the quantum Bruhat graph\nby Arghya Sadhukhan (Maryland) a
s part of UCSD number theory seminar\n\nLecture held in APM 6402 and onlin
e.\n\nAbstract\nThe study of affine Deligne-Lusztig varieties (ADLVs) $X_w
(b)$ and their certain union $X(\\mu\,b)$ has been crucial in understandin
g mod-$p$ reduction of Shimura varieties\; for instance\, precise informat
ion about the connected components of ADLVs (in the hyperspecial level) ha
s proved to be useful in Kisin's proof of the Langlands-Rapoport conjectur
e. On the other hand\, first introduced in the context of enumerative geom
etry to describe the quantum cohomology ring of complex flag varieties\, q
uantum Bruhat graphs have found recent applications in solving certain pro
blems on the ADLVs. I will survey such developments and report on my work
surrounding a dimension formula for $X(\\mu\,b)$ in the quasi-split case\,
as well as some partial description of the dimension and top-dimensional
irreducible components in the non quasi-split case.\n\npre-talk at 1:20pm\
n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Vallieres (CSU Chico/UC San Diego)
DTSTART;VALUE=DATE-TIME:20230216T220000Z
DTEND;VALUE=DATE-TIME:20230216T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/87
DESCRIPTION:Title: Iwasawa theory and graph theory\nby Daniel Vallieres (CSU Chico/UC S
an Diego) as part of UCSD number theory seminar\n\nLecture held in APM 640
2 and online.\n\nAbstract\nAnalogies between number theory and graph theor
y have been studied for quite some times now. During the past few years\,
it has been observed in particular that there is an analogy between class
ical Iwasawa theory and some phenomena in graph theory. In this talk\, we
will explain this analogy and present some of the results that have been
obtained so far in this area.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gilyoung Cheong (UC Irvine)
DTSTART;VALUE=DATE-TIME:20230202T220000Z
DTEND;VALUE=DATE-TIME:20230202T230000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/88
DESCRIPTION:Title: Polynomial equations for matrices over integers modulo a prime power and
the cokernel of a random matrix\nby Gilyoung Cheong (UC Irvine) as pa
rt of UCSD number theory seminar\n\nLecture held in APM 6402 and online.\n
\nAbstract\nOver a commutative ring of finite cardinality\, how many $n \\
times n$ matrices satisfy a polynomial equation? In this talk\, I will exp
lain how to solve this question when the ring is given by integers modulo
a prime power and the polynomial is square-free modulo the prime. Then I w
ill discuss how this question is related to the distribution of the cokern
el of a random matrix and the Cohen--Lenstra heuristics. This is joint wor
k with Yunqi Liang and Michael Strand\, as a result of a summer undergradu
ate research I mentored.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keegan Ryan (UC San Diego)
DTSTART;VALUE=DATE-TIME:20230420T210000Z
DTEND;VALUE=DATE-TIME:20230420T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/89
DESCRIPTION:Title: Fast Practical Lattice Reduction through Iterated Compression\nby Ke
egan Ryan (UC San Diego) as part of UCSD number theory seminar\n\nLecture
held in APM 6402 and online.\n\nAbstract\nWe introduce a new lattice basis
reduction algorithm with approximation guarantees analogous to the LLL al
gorithm and practical performance that far exceeds the current state of th
e art. We achieve these results by iteratively applying precision manageme
nt techniques within a recursive algorithm structure and show the stabilit
y of this approach. We analyze the asymptotic behavior of our algorithm\,
and show that the heuristic running time is $O(n^{\\omega}(C+n)^{1+\\varep
silon})$ for lattices of dimension $n$\, $\\omega\\in (2\,3]$ bounding the
cost of size reduction\, matrix multiplication\, and QR factorization\, a
nd $C$ bounding the log of the condition number of the input basis $B$. Th
is yields a running time of $O\\left(n^\\omega (p + n)^{1 + \\varepsilon}\
\right)$ for precision $p = O(\\log \\|B\\|_{max})$ in common applications
. Our algorithm is fully practical\, and we have published our implementat
ion. We experimentally validate our heuristic\, give extensive benchmarks
against numerous classes of cryptographic lattices\, and show that our alg
orithm significantly outperforms existing implementations.\n\npre-talk at
1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hanlin Cai (Utah)
DTSTART;VALUE=DATE-TIME:20230504T210000Z
DTEND;VALUE=DATE-TIME:20230504T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/91
DESCRIPTION:Title: Perfectoid signature and local étale fundamental group\nby Hanlin C
ai (Utah) as part of UCSD number theory seminar\n\nLecture held in APM 640
2 and online.\n\nAbstract\nIn this talk I'll talk about a (perfectoid) mix
ed characteristic version of F-signature and Hilbert-Kunz multiplicity by
utilizing the perfectoidization functor of Bhatt-Scholze and Faltings' nor
malized length. These definitions coincide with the classical theory in eq
ual characteristic. Moreover\, perfectoid signature detects BCM regularity
and transforms similarly to F-signature or normalized volume under quasi-
étale maps. As a consequence\, we can prove that BCM-regular rings have f
inite local étale fundamental group and torsion part of their divisor cla
ss groups. This is joint work with Seungsu Lee\, Linquan Ma\, Karl Schwede
and Kevin Tucker.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Kobin (Emory)
DTSTART;VALUE=DATE-TIME:20230518T210000Z
DTEND;VALUE=DATE-TIME:20230518T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/92
DESCRIPTION:Title: Categorifying zeta and L-functions\nby Andrew Kobin (Emory) as part
of UCSD number theory seminar\n\nLecture held in APM 6402 and online.\n\nA
bstract\nZeta and L-functions are ubiquitous in modern number theory. Whil
e some work in the past has brought homotopical methods into the theory of
zeta functions\, there is in fact a lesser-known zeta function that is na
tive to homotopy theory. Namely\, every suitably finite decomposition spac
e (aka 2-Segal space) admits an abstract zeta function as an element of it
s incidence algebra. In this talk\, I will show how many 'classical' zeta
functions in number theory and algebraic geometry can be realized in this
homotopical framework. I will also discuss work in progress towards a cate
gorification of motivic zeta and L-functions.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samit Dasgupta (Duke)
DTSTART;VALUE=DATE-TIME:20230511T210000Z
DTEND;VALUE=DATE-TIME:20230511T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/93
DESCRIPTION:Title: Ribet’s Lemma\, the Brumer-Stark Conjecture\, and the Main Conjecture<
/a>\nby Samit Dasgupta (Duke) as part of UCSD number theory seminar\n\nLec
ture held in APM 6402 and online.\n\nAbstract\nIn 1976\, Ken Ribet used mo
dular techniques to prove an important relationship between class groups o
f cyclotomic fields and special values of the zeta function. Ribet’s me
thod was generalized to prove the Iwasawa Main Conjecture for odd primes p
by Mazur-Wiles over Q and by Wiles over arbitrary totally real fields. \
n\nCentral to Ribet’s technique is the construction of a nontrivial exte
nsion of one Galois character by another\, given a Galois representation s
atisfying certain properties. Throughout the literature\, when working in
tegrally at p\, one finds the assumption that the two characters are not c
ongruent mod p. For instance\, in Wiles’ proof of the Main Conjecture\,
it is assumed that p is odd precisely because the relevant characters mig
ht be congruent modulo 2\, though they are necessarily distinct modulo any
odd prime.\n\nIn this talk I will present a proof of Ribet’s Lemma in t
he case that the characters are residually indistinguishable. As arithmet
ic applications\, one obtains a proof of the Iwasawa Main Conjecture for t
otally real fields at p=2. Moreover\, we complete the proof of the Brumer
-Stark conjecture by handling the localization at p=2\, building on joint
work with Mahesh Kakde for odd p. Our results yield the full Equivariant
Tamagawa Number conjecture for the minus part of the Tate motive associate
d to a CM abelian extension of a totally real field\, which has many impor
tant corollaries.\n\nThis is joint work with Mahesh Kakde\, Jesse Silliman
\, and Jiuya Wang.\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nha Truong (Hawaii)
DTSTART;VALUE=DATE-TIME:20230406T210000Z
DTEND;VALUE=DATE-TIME:20230406T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/94
DESCRIPTION:Title: Slopes of modular forms and geometry of eigencurves\nby Nha Truong (
Hawaii) as part of UCSD number theory seminar\n\nLecture held in APM 6402
and online.\n\nAbstract\nThe slopes of modular forms are the $p$-adic valu
ations of the eigenvalues of the Hecke operators $T_p$. The study of slope
s plays an important role in understanding the geometry of the eigencurves
\, introduced by Coleman and Mazur. \n\nThe study of the slope began in th
e 1990s when Gouvea and Mazur computed many numerical data and made severa
l interesting conjectures. Later\, Buzzard\, Calegari\, and other people m
ade more precise conjectures by studying the space of overconvergent modul
ar forms. Recently\, Bergdall and Pollack introduced the ghost conjecture
that unifies the previous conjectures in most cases. The ghost conjecture
states that the slope can be predicted by an explicitly defined power seri
es. We prove the ghost conjecture under a certain mild technical condition
. In the pre-talk\, I will explain an example in the quaternionic setting
which was used as a testing ground for the proof. \nThis is joint work wit
h Ruochuan Liu\, Liang Xiao\, and Bin Zhao.\n\npre-talk at 1:30pm (note un
usual time)\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Somnath Jha (IIT Kanpur)
DTSTART;VALUE=DATE-TIME:20230427T210000Z
DTEND;VALUE=DATE-TIME:20230427T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/95
DESCRIPTION:Title: Rational cube sum problem\nby Somnath Jha (IIT Kanpur) as part of UC
SD number theory seminar\n\nLecture held in APM 6402 and online.\n\nAbstra
ct\nThe classical Diophantine problem of determining which integers can b
e written as a sum of two rational cubes has a long history\; it includes
works of Sylvester\, Selmer\, Satgé\, Leiman and the recent work of Alp
öge-Bhargava-Shnidman-Burungale-Skinner. In this talk\, we will use Se
lmer groups of elliptic curves and integral binary cubic forms to study so
me cases of the rational cube sum problem. This talk is based on joint w
orks with D. Majumdar\, P. Shingavekar and B. Sury.\n\npre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Bucur (UC San Diego)
DTSTART;VALUE=DATE-TIME:20230413T210000Z
DTEND;VALUE=DATE-TIME:20230413T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/96
DESCRIPTION:Title: Counting $D_4$ quartic extensions of a number field ordered by discrimin
ant\nby Alina Bucur (UC San Diego) as part of UCSD number theory semin
ar\n\nLecture held in APM 6402 and online.\n\nAbstract\nA guiding question
in number theory\, specifically in arithmetic statistics\, is counting nu
mber fields of fixed degree and Galois group as their discriminants grow t
o infinity. We will discuss the history of this question and take a close
r look at the story in the case of quartic fields. In joint work with Flor
ea\, Serrano Lopez\, and Varma\, we extend and make explicit the counts of
extensions of an arbitrary number field that was done over the rationals
by Cohen\, Diaz y Diaz\, and Olivier.\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Catherine Hsu (Swarthmore College)
DTSTART;VALUE=DATE-TIME:20230601T210000Z
DTEND;VALUE=DATE-TIME:20230601T220000Z
DTSTAMP;VALUE=DATE-TIME:20230610T172907Z
UID:UCSD_NTS/97
DESCRIPTION:Title: Explicit non-Gorenstein R=T via rank bounds\nby Catherine Hsu (Swart
hmore College) as part of UCSD number theory seminar\n\nLecture held in AP
M 6402 and online.\n\nAbstract\nIn his seminal work on modular curves and
the Eisenstein ideal\, Mazur studied the existence of congruences between
certain Eisenstein series and newforms\, proving that Eisenstein ideals as
sociated to weight 2 cusp forms of prime level are locally principal. In t
his talk\, we'll explore generalizations of Mazur's result to squarefree l
evel\, focusing on recent work\, joint with P. Wake and C. Wang-Erickson\,
about a non-optimal level N that is the product of two distinct primes an
d where the Galois deformation ring is not expected to be Gorenstein. Firs
t\, we will outline a Galois-theoretic criterion for the deformation ring
to be as small as possible\, and when this criterion is satisfied\, deduce
an R=T theorem. Then we'll discuss some of the techniques required to com
putationally verify the criterion.\n\nPre-talk\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/97/
END:VEVENT
END:VCALENDAR