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BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard University)
DTSTART:20210922T221000Z
DTEND:20210922T231000Z
DTSTAMP:20260404T094940Z
UID:number_theory/1
DESCRIPTION:Title: Heegner points and patched Euler systems in anticyclotomic Iw
asawa theory\nby Naomi Sweeting (Harvard University) as part of UBC Nu
mber theory seminar\n\n\nAbstract\nThis talk will report on recent work pr
oving new cases of the\nHeegner Point Main Conjecture of Perrin-Riou. I'll
explain the statement of\nthe conjecture and the method of patched bipart
ite Euler systems used in\nthe proof. This method reduces the HPMC to a ma
in conjecture of Bertolini\nand Darmon "at infinite level"\, which can be
resolved using the work of\nSkinner-Urban along with a deformation-theoret
ic input following methods of\nFakhruddin-Khare-Patrikis. One consequence
of the results is an improved\np-converse theorem to the work of Gross-Zag
ier and Kolyvagin: p-Selmer rank\none implies analytic rank one.\n\nPlease
sign up for the talk using the link https://ubc.zoom.us/meeting/register/
u5Yrfu2sqTkoH9AqIzq7m7896a2yg2A6BlSe and the zoom link will be sent to you
r mailing address.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lea Beneish (UC Berkeley)
DTSTART:20210915T220000Z
DTEND:20210915T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/2
DESCRIPTION:Title: Fields generated by points on superelliptic curves\nby Le
a Beneish (UC Berkeley) as part of UBC Number theory seminar\n\n\nAbstract
\nWe give an asymptotic lower bound on the number of field\nextensions gen
erated by algebraic points on superelliptic curves over\n$\\mathbb{Q}$ wit
h fixed degree $n$\, discriminant bounded by $X$\, and Galois\nclosure $S_
n$. For $C$ a fixed curve given by an affine equation $y^m =\nf(x)$ where
$m \\geq 2$ and $deg f(x) = d \\geq m$\, we find that for all\ndegrees $n$
divisible by $gcd(m\, d)$ and sufficiently large\, the number of\nsuch fi
elds is asymptotically bounded below by $X^{c_n}$ \, where $c_n$ goes to\n
$1/m^2$ as $n$ goes to $\\infty$. This bound is determined explicitly by\n
parameterizing $x$ and $y$ by rational functions\, counting specialization
s\,\nand accounting for multiplicity. We then give geometric heuristics\ns
uggesting that for $n$ not divisible by $gcd(m\, d)$\, degree $n$ points\n
may be less abundant than those for which $n$ is divisible by $gcd(m\, d)$
.\nNamely\, we discuss the obvious geometric sources from which we expect
to\nfind points on $C$ and discuss the relationship between these sources
and\nour parametrization. When one a priori has a point on $C$ of degree n
ot\ndivisible by $gcd(m\, d)$\, we argue that a similar counting argument\
napplies. As a proof of concept we show in the case that $C$ has a rationa
l\npoint that our methods can be extended to bound the number of fields\ng
enerated by a degree $n$ point of $C$\, regardless of divisibility of $n$\
nby $gcd(m\, d)$. This talk is based on joint work with Christopher Keyes.
\n\nPlease sign up for the talk using the link https://ubc.zoom.us/meeting
/register/u5Yrfu2sqTkoH9AqIzq7m7896a2yg2A6BlSe and the zoom link will be s
ent to your mailing address.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anwesh Ray (University of British Columbia)
DTSTART:20211013T220000Z
DTEND:20211013T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/3
DESCRIPTION:Title: Arithmetic statistics and diophantine stability for elliptic
curves\nby Anwesh Ray (University of British Columbia) as part of UBC
Number theory seminar\n\n\nAbstract\nIn 2017\, B. Mazur and K. Rubin intro
duced the notion of diophantine stability for a variety defined over a num
ber field. Given an elliptic curve E defined over the rationals and a prim
e number p\, E is said to be diophantine stable at p if there are abundant
ly many p-cyclic extensions $L/\\mathbb{Q}$ such that $E(L)=E(\\mathbb{Q})
$. In particular\, this means that given any integer $n>0$\, there are inf
initely many cyclic extensions with Galois group $\\mathbb{Z}/p^n\\mathbb{
Z}$\, such that $E(L)=E(\\mathbb{Q})$. It follows from more general result
s of Mazur-Rubin that $E$ is diophantine stable at a positive density set
of primes p.\n\nIn this talk\, I will discuss diophantine stability of ave
rage for pairs $(E\,p)$\, where $E$ is a non-CM elliptic curve and $p\\geq
11$ is a prime number at which $E$ has good ordinary reduction. First\, I
will fix the elliptic curve and vary the prime. In this context\, it is s
hown that diophantine stability is a consequence of certain properties of
Selmer groups studied in Iwasawa theory. Statistics for Iwasawa invariants
were studied recently (in joint work with collaborators). As an applicati
on\, one shows that if the Mordell Weil rank of E is zero\, then\, $E$ is
diophantine stable at $100\\%$ of primes $p$. One also shows that standard
conjectures (like rank distribution) imply that for any prime $p\\geq 11$
\, a positive density set of elliptic curves (ordered by height) is diopha
ntine stable at $p$. I will also talk about related results for stability
and growth of the p-primary part of the Tate-Shafarevich group in cyclic p
-extensions.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumya Sankar (Ohio State University)
DTSTART:20211020T220000Z
DTEND:20211020T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/4
DESCRIPTION:Title: Counting elliptic curves with rational N-isogeny\nby Soum
ya Sankar (Ohio State University) as part of UBC Number theory seminar\n\n
\nAbstract\nThe classical problem of counting elliptic curves with a ratio
nal\nN-isogeny can be phrased in terms of counting rational points on cert
ain moduli\nstacks of elliptic curves. Counting points on stacks poses var
ious challenges\,\nand I will discuss these along with a few ways to overc
ome them. I will also\ntalk about the theory of heights on stacks develope
d in recent work of\nEllenberg\, Satriano and Zureick-Brown and use it to
count elliptic curves with\nan N-isogeny for certain N. The talk assumes n
o prior knowledge of stacks and\nis based on joint work with Brandon Bogge
ss.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UC Santa Barbara)
DTSTART:20210929T221000Z
DTEND:20210929T231000Z
DTSTAMP:20260404T094940Z
UID:number_theory/5
DESCRIPTION:Title: Heegner points and generalised Kato classes\nby Francesc
Castella (UC Santa Barbara) as part of UBC Number theory seminar\n\n\nAbst
ract\nFor an elliptic curve $E/\\mathbb{Q}$ and a fixed prime $p$\, a\ncel
ebrated "$p$-converse" to a theorem of Kolyvagin takes the form of the\nim
plication: If the $p^\\infty$ Selmer group of $E$ has\n$\\mathbb{Z}_p$-cor
ank one\, then a certain Heegner is non-torsion. The\nGross-Zagier formula
then allows one to conclude that $E$ has analytic rank\none. Following th
e pioneering work of Skinner and Wei Zhang\, a growing\nnumber of results
are known in the direction of this $p$-converse. In this\ntalk\, I'll desc
ribe the proof of a result in the same spirit for elliptic\ncurves of rank
two\, in which Heegner points are replaced by certain\ngeneralised Kato c
lasses introduced by Darmon and Rotger. The talk is based\non joint work w
ith M.-L. Hsieh.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Artane Siad (IAS/Princeton)
DTSTART:20211027T220000Z
DTEND:20211027T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/6
DESCRIPTION:Title: Monogenic fields with odd class number\nby Artane Siad (I
AS/Princeton) as part of UBC Number theory seminar\n\n\nAbstract\nIn this
talk\, we prove an upper bound on the average number of 2-torsion elements
in the class group of monogenised fields of any degree $n\\geq 3$\, and\,
conditional on a widely expected tail estimate\, compute this average exa
ctly. As an application\, we show that there are infinitely many number fi
elds with odd class number in any even degree and signature. Time permitti
ng\, we will also discuss extensions of this result to orders and to the r
elative setting.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ellen Eischen (U Oregon)
DTSTART:20211201T230000Z
DTEND:20211202T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/7
DESCRIPTION:Title: p-adic aspects of L-functions\, with a view toward Spin L-fun
ctions\nby Ellen Eischen (U Oregon) as part of UBC Number theory semin
ar\n\n\nAbstract\nThe study of p-adic properties of values of L-functions
dates back to Kummer's study of congruences between values of the Riemann
zeta function at negative odd\nintegers\, as part of his attempt to unders
tand class numbers of cyclotomic extensions.\nAfter Kummer's ideas largely
lay dormant for over a half century\, Iwasawa's conjectures\nabout the me
aning of p-adic L-functions led to renewed interest\, and Serre's discover
y of\np-adic modular forms opened up a new approach to studying congruence
s between values\nof L-functions\, forming the foundation for continued de
velopments today.\n\nWith a viewpoint that encompasses several settings\,
including modular forms (on $GL_2$)\nand automorphic forms on higher rank
groups\, I will introduce p-adic L-functions and a\nrecipe for constructin
g them\, which relies partly on properties of Fourier coefficients of modu
lar (and automorphic) forms. Along the way\, I will introduce several rece
nt developments\nand put them in the context of constructions of Serre\, K
atz\, and Hida. As an example of a\nrecent application of these ideas\, I
will discuss the results of a paper-in-preparation\, joint\nwith G. Rosso
and S. Shah\, on p-adic Spin L-functions of ordinary cuspidal automorphic\
nrepresentations of $GSp_6$ associated to Siegel modular forms.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rylan Gajek-Leonard (UMass Amherst)
DTSTART:20211006T220000Z
DTEND:20211006T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/8
DESCRIPTION:Title: Iwasawa Invariants of Modular Forms with $a_p=0$\nby Ryla
n Gajek-Leonard (UMass Amherst) as part of UBC Number theory seminar\n\n\n
Abstract\nMazur-Tate elements provide a convenient method to study the\nan
alytic Iwasawa theory of p-nonordinary modular forms\, where the\nassociat
ed p-adic L-functions tend to have unbounded coefficients. The\nIwasawa in
variants of Mazur-Tate elements are well-understood in the case\nof weight
2 modular forms\, where they can be related to the growth of\np-Selmer gr
oups and decompositions of the p-adic L-function. At higher\nweights\, les
s is known. By constructing certain lifts to the full Iwasawa\nalgebra\, w
e compute the Iwasawa invariants of Mazur-Tate elements for\nhigher weight
modular forms with $a_p=0$ in terms of the plus/minus\ninvariants of the
p-adic L-function. Combined with results of\nPollack-Weston\, this forces
a relation between the plus/minus invariants\nat weights 2 and p+1.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ethan White & Chi Hoi Yip (University of British Columbia)
DTSTART:20211117T230000Z
DTEND:20211118T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/9
DESCRIPTION:Title: The number of directions determined by a Cartesian product in
the affine Galois plane\nby Ethan White & Chi Hoi Yip (University of
British Columbia) as part of UBC Number theory seminar\n\n\nAbstract\nThe
directions determined by a subset $U \\subset AG(2\,p)$ is the set of slop
es\nformed by pairs of points from $U$. For $U = A \\times B$\, a Cartesia
n product\,\nwe give a new lower bound on the number of directions determi
ned by $U$.\nCombining this result with estimates on exponential sums\, we
make progress on\nthe Paley graph conjecture (a double character sum esti
mate).\n\nWhen $A=B$ is an arithmetic progression\, we give an asymptotic
formula for the\nnumber of directions. Our method involves computing an as
ymptotic formula for\nthe number of solutions to the Diophantine equation
$ad+bc = p$.\n\nJoint work with Daniel Di Benedetto\, Greg Martin\, and Jo
zsef Solymosi.\n\nSpeakers: Ethan White (UBC) homepage: https://personal.m
ath.ubc.ca/~epwhite/ \n\nChi Hoi Yip (UBC) homepage: https://sites.google.
com/view/kyle-chi-hoi-yip/home \n\nSince the site only allows for one home
page link\, I went with that of the second named speaker (based on the las
t named speaker given first priority principle).\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalia Garcia-Fritz (Pontificia Universidad Católica de Chile)
DTSTART:20211103T220000Z
DTEND:20211103T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/10
DESCRIPTION:Title: Approaching Hilbert's Tenth problem for rings of integers of
number fields through Iwasawa theory\nby Natalia Garcia-Fritz (Pontif
icia Universidad Católica de Chile) as part of UBC Number theory seminar\
n\n\nAbstract\nAfter the solution by Davis\, Putnam\, Robinson and Matiyas
evich of Hilbert's\nTenth problem for the integers\, a natural extension t
hat remains mostly open is the analogue for rings of integers of number fi
elds. Several cases were proved in the seventies\nand eighties by Denef\,
Lipshitz\, Pheidas\, Shlapentokh\, Videla and Shapiro\, but after that\npo
int there has been a long hiatus on unconditional results. Most recently\,
elliptic curve\ncriteria by Poonen\, Cornelissen-Pheidas-Zahidi and Shlap
entokh have led to a complete\nsolution under standard arithmetic conjectu
res\, thanks to the work of Mazur-Rubin and\nMurty-Pasten. In this talk\,
I will present some unconditional cases proved in joint work\nwith Hector
Pasten. The proof is based on the elliptic curve criteria\, and it uses re
cent\ntechniques from Iwasawa theory and Heegner points.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashay Burungale (Caltech/UT Austin)
DTSTART:20211124T231000Z
DTEND:20211125T001000Z
DTSTAMP:20260404T094940Z
UID:number_theory/11
DESCRIPTION:Title: A conjecture of Rubin\nby Ashay Burungale (Caltech/UT Au
stin) as part of UBC Number theory seminar\n\n\nAbstract\nIn 1987\, with a
n eye towards anticyclotomic Iwasawa theory of CM\nelliptic curves at iner
t primes\, Rubin proposed a basic conjecture on the\nstructure of Iwasawa
module of local units over anticyclotomic extensions\nof the unramified qu
adratic extension of $\\mathbb{Q}_p$. The talk will report on\nrecent proo
f of Rubin's conjecture and some of subsequent developments\n(joint with S
. Kobayashi and K. Ota).\n\nTo accommodate a special request\, we will be
starting tomorrow's seminar at\n*3:10PM.*\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prajeet Bajpai (UBC)
DTSTART:20211208T230000Z
DTEND:20211209T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/12
DESCRIPTION:Title: Effective Methods for Norm-Form Equations\nby Prajeet Ba
jpai (UBC) as part of UBC Number theory seminar\n\n\nAbstract\nLet $\\alph
a_1\,\\ldots\,\\alpha_k$ be linearly independent elements\nof a number fie
ld $K$ of degree $n \\ge k$\, and let $m$ be an integer. The\nequation $\\
mathrm{Norm}_{K/\\Q} (x_1\\alpha_1 + \\cdots + x_k\\alpha_k) = m$\,\nto be
solved in integers\, is called a `norm-form equation'. The case of\nbinar
y forms was solved by Thue in 1909\, and the general case was resolved\nby
Schmidt in 1971 through his Subspace Theorem generalizing the work of\nTh
ue-Siegel-Roth. Unfortunately these results are ineffective\, and do not\n
provide any means of determining a bound on the height of exceptional\nsol
utions-- in particular\, they do not allow us to determine a complete\nlis
t of solutions for even a single norm-form equation.\n\nBaker's theorem on
linear forms in logarithms gave an effective version of\nThue's result fo
r binary forms\, and Vojta in his PhD thesis was able extend\nthis effecti
vity to three-variable norm-form equations under the assumption\nthat $K$
is totally complex and Galois. In this talk we discuss effective\nresoluti
on for certain norm-form equations in four and five variables\,\nextending
the work of Vojta. In particular\, we completely and effectively\nresolve
the question of norm-form equations over totally complex Galois\nsextic f
ields. The results are motivated by joint work with Mike Bennett.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Hartmann (UPenn)
DTSTART:20211215T230000Z
DTEND:20211216T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/13
DESCRIPTION:Title: Local-global principles for linear algebraic groups over ari
thmetic function fields\nby Julia Hartmann (UPenn) as part of UBC Numb
er theory seminar\n\n\nAbstract\nArithmetic function fields are one variab
le function fields over complete\ndiscretely valued fields. They naturally
admit several collections of\noverfields with respect to which one can st
udy local-global principles. The\ntalk will concern such local-global prin
ciples for torsors under linear\nalgebraic groups\, as well as their obstr
uctions. (Joint work with\nJ.L.-Colliot-Thélène\, D. Harbater\, D. Krash
en\, R. Parimala\, and V. Suresh.)\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbey Bourdon (Wake Forest University)
DTSTART:20211220T230000Z
DTEND:20211221T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/14
DESCRIPTION:Title: Torsion Subgroups of CM Elliptic Curves in Degree 2p\nby
Abbey Bourdon (Wake Forest University) as part of UBC Number theory semin
ar\n\n\nAbstract\nA common classification problem is to identify the group
s which arise as\nthe torsion subgroup of an elliptic curve defined over a
ny number field of\na fixed degree. That only finitely many such groups oc
cur in this context\nis a consequence of Merel's Uniform Boundedness Theor
em. However\, for\ncertain families of elliptic curves--such as those with
complex\nmultiplication (CM)--recent advances have allowed us to move bey
ond a\nfixed-degree classification to glimpse the behavior of torsion poin
ts over\ninfinitely many degrees of a restricted form. In this talk\, I wi
ll discuss\nrecent work with Holly Paige Chaos which characterizes the gro
ups that\narise as torsion subgroups of CM elliptic curves defined over nu
mber fields\nof degree 2p where p is prime. Here\, a classification in the
strongest\nsense is tied to determining whether there exist infinitely ma
ny Sophie\nGermain primes.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Heidi Goodson (Brooklyn College)
DTSTART:20220112T230000Z
DTEND:20220113T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/15
DESCRIPTION:Title: Sato-Tate Groups in Dimension Greater than 3\nby Heidi G
oodson (Brooklyn College) as part of UBC Number theory seminar\n\n\nAbstra
ct\nThe focus of this talk is on Sato-Tate groups of abelian varieties --\
ncompact groups predicted to determine the limiting distributions of local
zeta\nfunctions. In recent years\, complete classifications of Sato-Tate
groups in\ndimensions 1\, 2\, and 3 have been given\, but there are obstac
les to providing\nclassifications in higher dimensions. In this talk\, I w
ill describe my recent\nwork on families of higher dimensional Jacobian va
rieties. This work is partly\njoint with Melissa Emory.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State University)
DTSTART:20220119T230000Z
DTEND:20220120T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/16
DESCRIPTION:Title: Eisenstein congruences and class groups of pure fields\n
by Preston Wake (Michigan State University) as part of UBC Number theory s
eminar\n\n\nAbstract\nLet $p$ and $N$ be primes and assume $N$ is $−1$ m
odulo $p$. Then the class number\nof $\\mathbb{Q}(N^{1/p})$ is divisible b
y $p$. I'll explain how to prove this using congruences between\nmodular f
orms. This is joint work with Jackie Lang\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Emory (University of Toronto)
DTSTART:20220126T230000Z
DTEND:20220127T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/17
DESCRIPTION:Title: Sato-Tate distributions for higher genus curves (TALK CANCEL
LED)\nby Melissa Emory (University of Toronto) as part of UBC Number t
heory seminar\n\n\nAbstract\nWe discuss work to determine Sato-Tate groups
for higher genus\ncurves. In so doing\, we detail an effective algorithm
that computes the\nidentity componenent of Sato-Tate groups.\nWe will als
o discuss open problems related to this work and graduate\nstudents are en
couraged to attend. This is joint work with H. Goodson and\nA.Peyrot.\n\n
Unfortunately\, we have to cancel today's talk.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matilde Lalin (Université de Montréal)
DTSTART:20220209T230000Z
DTEND:20220210T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/18
DESCRIPTION:Title: On the Northcott property of $L$-functions over function fie
lds\nby Matilde Lalin (Université de Montréal) as part of UBC Number
theory seminar\n\n\nAbstract\nThe Northcott property implies that a set o
f algebraic numbers with bounded height and bounded degree must be finite.
Pazuki and Pengo introduced a variant of the Northcott property for numbe
r fields using special values of the Dedekind zeta function to measure the
height. We consider this question for global function fields with constan
t fields $\\mathbb{F}_q$. This is joint work with Xavier Genereux and Wanl
in Li.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly Isham (Colgate University)
DTSTART:20220202T230000Z
DTEND:20220203T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/19
DESCRIPTION:Title: Zeta functions and asymptotics related to subrings in $\\mat
hbb{Z}^n$\nby Kelly Isham (Colgate University) as part of UBC Number t
heory seminar\n\n\nAbstract\nWe can define a zeta function of a group (or
ring) to be the Dirichlet series associated to the sequence that counts th
e number of subgroups (or subrings) of a given index. The subgroup zeta fu
nction over $\\mathbb{Z}^n$ is well-understood\, as is the asymptotic grow
th of subgroups in $\\mathbb{Z}^n$. Much less is known about the subring z
eta function over $\\mathbb{Z}^n$ and the asymptotic growth of subrings in
$\\mathbb{Z}^n$. In this talk\, we discuss the progress toward answering
this question and we give new lower bounds on the asymptotic growth of sub
rings in $\\mathbb{Z}^n$. We also define a similar zeta function correspon
ding to subrings of corank at most k in $\\mathbb{Z}^n$. While the proport
ion of subgroups in $\\mathbb{Z}^n$ of corank $k$ is positive for each $k$
\, we show this is not the case for subrings in $\\mathbb{Z}^n$ of corank
$k$ when $n$ is sufficiently larger than $k$. Lastly\, we make connections
to orders in number fields. Part of this work is joint with Nathan Kaplan
.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar (University of Wisconsin Madison)
DTSTART:20220316T220000Z
DTEND:20220316T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/20
DESCRIPTION:Title: Canonical heights and the Andre-Oort conjecture\nby Anan
th Shankar (University of Wisconsin Madison) as part of UBC Number theory
seminar\n\n\nAbstract\nLet S be a Shimura variety. The Andre-Oort conjectu
re posits that the Zariski\nclosure of special points must be a sub-Shimur
a subvariety of S. The Andre-Oort\nconjecture for $A_g$ (the moduli space
of principally polarized Abelian\nvarieties) and therefore its sub-Shimura
varieties was proved by Jacob\nTsimerman. However\, this conjecture was u
nknown for Shimura varieties without a\nmoduli interpretation. I will desc
ribe joint work with Jonathan Pila and Jacob\nTsimerman (with an appendix
by Esnault-Groechenig) where we prove the Andre-Oort conjecture in full ge
nerality.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cedric Dion (Univ. Laval)
DTSTART:20220309T230000Z
DTEND:20220310T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/21
DESCRIPTION:Title: Iwasawa theory for supersingular abelian varieties\nby C
edric Dion (Univ. Laval) as part of UBC Number theory seminar\n\n\nAbstrac
t\nFix an odd prime number p. Let K be a quadratic imaginary field where p
\nsplits. Let A be an abelian variety de\nned over K with good supersingul
ar reduction at\nboth primes above p. In this talk\, we investigate some a
spect of the Iwasawa theory for A\nover the $\\mathbb{Z}_p^2$-extension of
K. We begin by giving an overview of the relevant Selmer groups\nbuilding
on the work of Büyükboduk and Lei. We then show that the Mordeil-Weil r
ank of\nA along this $\\mathbb{Z}_p^2$-extension grows like a function whi
ch is $O(p^n\n)$ (joint with J. Ray). Finally\,\nusing the recently develo
ped theory of gamma-systems\, we prove an algebraic functional\nequation i
nvolving the Pontryagin dual of our Selmer groups.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabella Negrini (Mcgill University)
DTSTART:20220216T230000Z
DTEND:20220217T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/22
DESCRIPTION:Title: A Shimura-Shintani correspondence for rigid analytic cocycle
s\nby Isabella Negrini (Mcgill University) as part of UBC Number theor
y seminar\n\n\nAbstract\nIn their paper Singular moduli for real quadratic
fields: a rigid\n analytic approach\, Darmon and Vonk introduced rigid me
romorphic cocycles\,\n i.e. elements of $H^1(SL_2(\\mathbb{Z}[1/p])\, M^x)
$ where $M^x$ is the multiplicative\n group of rigid meromorphic functions
on the p-adic upper-half plane.\n Their values at RM points belong to nar
row ring class fields of real\n quadratic fiends and behave analogously to
CM values of modular functions\n on $SL_2(\\mathbb{Z})\\backslash H$. In
this talk I will present some progress towards\n developing a Shimura-Shin
tani correspondence in this setting.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liyang Yang (Princeton)
DTSTART:20220302T230000Z
DTEND:20220303T000000Z
DTSTAMP:20260404T094940Z
UID:number_theory/23
DESCRIPTION:Title: Relative Trace Formula and Average Central L-values on $U(3)
× U(2)$\nby Liyang Yang (Princeton) as part of UBC Number theory sem
inar\n\n\nAbstract\nIn this talk\, we will introduce an explicit relative
trace formula to study central\nL-values on $U(3)×U(2)$. In conjunction w
ith computations of local factors in Ichino-Ikeda\nformulas for Bessel per
iods\, we obtain some important properties of these central L-values\nover
certain family: the first moment\, nonvanishing and subconvexity in the l
evel aspect.\nThis is joint work with Philippe Michel and Dinakar Ramakris
hnan.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siddhi Pathak (Chennai Mathematical Institute)
DTSTART:20220324T173000Z
DTEND:20220324T183000Z
DTSTAMP:20260404T094940Z
UID:number_theory/24
DESCRIPTION:Title: Special values of L-functions - a classical approach\nby
Siddhi Pathak (Chennai Mathematical Institute) as part of UBC Number theo
ry seminar\n\n\nAbstract\nIn 1734\, Euler observed that the values of the
Riemann zeta-function\nat even positive integers are rational multiples of
powers of $ \\pi $. However\,\nthe odd zeta-values remain a mystery to th
is day. In fact\, it is widely\nbelieved that the odd zeta-values do not s
atisfy any polynomial relation with $\n\\pi $ over the rational numbers. A
lmost three centuries after Euler\, several\ndifferent perspectives have e
merged to study the general case of special values\nof L-functions. In thi
s talk\, we discuss a more classical approach and describe\nrecent progres
s on related conjectures by Chowla\, Erdos and Milnor.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Heejong Lee (University of Toronto)
DTSTART:20220330T220000Z
DTEND:20220330T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/25
DESCRIPTION:Title: Mod p local-global compatibility for $GSp_4(\\mathbb{Q}_p)$
in the ordinary case\nby Heejong Lee (University of Toronto) as part o
f UBC Number theory seminar\n\n\nAbstract\nThe conjectural mod p Langlands
program relates continuous mod p local Galois representations to smooth a
dmissible representation of p-adic groups (with coeficient\nfield of chara
cteristic p). Except for $GL_1$ and $GL_2(\\mathbb{Q}_p)$\, there is no kn
own construction of\nmod p Langlands correspondence. However\, it is possi
ble to construct a candidate using the\nspace of mod p automorphic forms a
nd the Taylor-Wiles method. It is not clear whether\nthis candidate\, cons
tructed by taking non-canonical choices of global data\, is determined\nby
the initial mod p local Galois representation.\nIn this talk\, we discuss
a question in reverse direction: can we recover the initial\nlocal Galois
representation from the smooth admissible representation of p-adic group\
nconstructed above? We will discuss the proof of this statement in the cas
e of $GSp_4(\\mathbb{Q}_p)$ and\nlocal Galois representations that are upp
er-triangular\, maximally non-split\, and generic.\nOne of the main ingred
ients in the proof is explicit Jantzen filtration of lattices in a certain
\ntame type. This is joint work with John Enns.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mishty Ray (University of Calgary)
DTSTART:20220406T220000Z
DTEND:20220406T230000Z
DTSTAMP:20260404T094940Z
UID:number_theory/26
DESCRIPTION:Title: Geometry of local Arthur packets for simple unramified Arthu
r parameters for $GL_n$\nby Mishty Ray (University of Calgary) as part
of UBC Number theory seminar\n\n\nAbstract\nThe local Langlands correspon
dence for a connected reductive p-adic group G partitions the set of equiv
alence classes of smooth irreducible representations of G(F) into L-packet
s using equivalence classes of Langlands parameters. Vogan's geometric per
spective gives us a moduli space of Langlands parameters\, and the corresp
ondence can be viewed as a relation between the set of equivalence classes
of smooth irreducible representations of G(F) and simple objects in the c
ategory of equivariant perverse sheaves on the moduli space of Langlands p
arameters that share a common infinitesimal parameter. This geometry gives
us the notion of an ABV-packet\, a set of smooth irreducible representati
ons of G(F)\, which conjecturally generalizes the notion of a local Arthur
packet - a local Arthur packet is conjecturally an ABV-packet. In this ta
lk\, we will look at Langlands parameters coming from simple Arthur parame
ters in the case of $GL_n$. We will explore the geometry of the moduli spa
ce of Langlands parameters using an example. We will see work in progress
towards proving that the local Arthur packet is the ABV-packet for this ca
se.\n
LOCATION:https://stable.researchseminars.org/talk/number_theory/26/
END:VEVENT
END:VCALENDAR