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BEGIN:VEVENT
SUMMARY:Naomi Sweeting (Harvard University)
DTSTART:20210922T221000Z
DTEND:20210922T231000Z
DTSTAMP:20260422T225924Z
UID:number_theory/1
DESCRIPTION:Title: Heegner points and patched Euler systems in anticyclotomic Iwasawa t
heory\nby Naomi Sweeting (Harvard University) as part of UBC Number th
eory seminar\n\n\nAbstract\nThis talk will report on recent work proving n
ew cases of the\nHeegner Point Main Conjecture of Perrin-Riou. I'll explai
n the statement of\nthe conjecture and the method of patched bipartite Eul
er systems used in\nthe proof. This method reduces the HPMC to a main conj
ecture of Bertolini\nand Darmon "at infinite level"\, which can be resolve
d using the work of\nSkinner-Urban along with a deformation-theoretic inpu
t following methods of\nFakhruddin-Khare-Patrikis. One consequence of the
results is an improved\np-converse theorem to the work of Gross-Zagier and
Kolyvagin: p-Selmer rank\none implies analytic rank one.\n\nPlease sign u
p for the talk using the link https://ubc.zoom.us/meeting/register/u5Yrfu2
sqTkoH9AqIzq7m7896a2yg2A6BlSe and the zoom link will be sent to your maili
ng address.\n
LOCATION:https://researchseminars.org/talk/number_theory/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lea Beneish (UC Berkeley)
DTSTART:20210915T220000Z
DTEND:20210915T230000Z
DTSTAMP:20260422T225924Z
UID:number_theory/2
DESCRIPTION:Title: Fields generated by points on superelliptic curves\nby Lea Benei
sh (UC Berkeley) as part of UBC Number theory seminar\n\n\nAbstract\nWe gi
ve an asymptotic lower bound on the number of field\nextensions generated
by algebraic points on superelliptic curves over\n$\\mathbb{Q}$ with 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 $m \\ge
q 2$ and $deg f(x) = d \\geq m$\, we find that for all\ndegrees $n$ divisi
ble by $gcd(m\, d)$ and sufficiently large\, the number of\nsuch fields 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\nparamet
erizing $x$ and $y$ by rational functions\, counting specializations\,\nan
d accounting for multiplicity. We then give geometric heuristics\nsuggesti
ng that for $n$ not divisible by $gcd(m\, d)$\, degree $n$ points\nmay be
less abundant than those for which $n$ is divisible by $gcd(m\, d)$.\nName
ly\, we discuss the obvious geometric sources from which we expect to\nfin
d points on $C$ and discuss the relationship between these sources and\nou
r parametrization. When one a priori has a point on $C$ of degree not\ndiv
isible by $gcd(m\, d)$\, we argue that a similar counting argument\napplie
s. As a proof of concept we show in the case that $C$ has a rational\npoin
t that our methods can be extended to bound the number of fields\ngenerate
d by a degree $n$ point of $C$\, regardless of divisibility of $n$\nby $gc
d(m\, d)$. This talk is based on joint work with Christopher Keyes.\n\nPle
ase sign up for the talk using the link https://ubc.zoom.us/meeting/regist
er/u5Yrfu2sqTkoH9AqIzq7m7896a2yg2A6BlSe and the zoom link will be sent to
your mailing address.\n
LOCATION:https://researchseminars.org/talk/number_theory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anwesh Ray (University of British Columbia)
DTSTART:20211013T220000Z
DTEND:20211013T230000Z
DTSTAMP:20260422T225924Z
UID:number_theory/3
DESCRIPTION:Title: Arithmetic statistics and diophantine stability for elliptic curves<
/a>\nby Anwesh Ray (University of British Columbia) as part of UBC Number
theory seminar\n\n\nAbstract\nIn 2017\, B. Mazur and K. Rubin introduced t
he notion of diophantine stability for a variety defined over a number fie
ld. Given an elliptic curve E defined over the rationals and a prime numbe
r p\, E is said to be diophantine stable at p if there are abundantly many
p-cyclic extensions $L/\\mathbb{Q}$ such that $E(L)=E(\\mathbb{Q})$. In p
articular\, this means that given any integer $n>0$\, there are infinitely
many cyclic extensions with Galois group $\\mathbb{Z}/p^n\\mathbb{Z}$\, s
uch that $E(L)=E(\\mathbb{Q})$. It follows from more general results of Ma
zur-Rubin that $E$ is diophantine stable at a positive density set of prim
es p.\n\nIn this talk\, I will discuss diophantine stability of average fo
r 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 f
ix the elliptic curve and vary the prime. In this context\, it is shown th
at diophantine stability is a consequence of certain properties of Selmer
groups studied in Iwasawa theory. Statistics for Iwasawa invariants were s
tudied recently (in joint work with collaborators). As an application\, on
e shows that if the Mordell Weil rank of E is zero\, then\, $E$ is diophan
tine stable at $100\\%$ of primes $p$. One also shows that standard conjec
tures (like rank distribution) imply that for any prime $p\\geq 11$\, a po
sitive density set of elliptic curves (ordered by height) is diophantine s
table at $p$. I will also talk about related results for stability and gro
wth of the p-primary part of the Tate-Shafarevich group in cyclic p-extens
ions.\n
LOCATION:https://researchseminars.org/talk/number_theory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumya Sankar (Ohio State University)
DTSTART:20211020T220000Z
DTEND:20211020T230000Z
DTSTAMP:20260422T225924Z
UID:number_theory/4
DESCRIPTION:Title: Counting elliptic curves with rational N-isogeny\nby Soumya Sank
ar (Ohio State University) as part of UBC Number theory seminar\n\n\nAbstr
act\nThe classical problem of counting elliptic curves with a rational\nN-
isogeny can be phrased in terms of counting rational points on certain mod
uli\nstacks of elliptic curves. Counting points on stacks poses various ch
allenges\,\nand I will discuss these along with a few ways to overcome the
m. I will also\ntalk about the theory of heights on stacks developed in re
cent work of\nEllenberg\, Satriano and Zureick-Brown and use it to count e
lliptic curves with\nan N-isogeny for certain N. The talk assumes no prior
knowledge of stacks and\nis based on joint work with Brandon Boggess.\n
LOCATION:https://researchseminars.org/talk/number_theory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UC Santa Barbara)
DTSTART:20210929T221000Z
DTEND:20210929T231000Z
DTSTAMP:20260422T225924Z
UID:number_theory/5
DESCRIPTION:Title: Heegner points and generalised Kato classes\nby Francesc Castell
a (UC Santa Barbara) as part of UBC Number theory seminar\n\n\nAbstract\nF
or an elliptic curve $E/\\mathbb{Q}$ and a fixed prime $p$\, a\ncelebrated
"$p$-converse" to a theorem of Kolyvagin takes the form of the\nimplicati
on: If the $p^\\infty$ Selmer group of $E$ has\n$\\mathbb{Z}_p$-corank one
\, then a certain Heegner is non-torsion. The\nGross-Zagier formula then a
llows one to conclude that $E$ has analytic rank\none. Following the pione
ering work of Skinner and Wei Zhang\, a growing\nnumber of results are kno
wn in the direction of this $p$-converse. In this\ntalk\, I'll describe th
e proof of a result in the same spirit for elliptic\ncurves of rank two\,
in which Heegner points are replaced by certain\ngeneralised Kato classes
introduced by Darmon and Rotger. The talk is based\non joint work with M.-
L. Hsieh.\n
LOCATION:https://researchseminars.org/talk/number_theory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Artane Siad (IAS/Princeton)
DTSTART:20211027T220000Z
DTEND:20211027T230000Z
DTSTAMP:20260422T225924Z
UID:number_theory/6
DESCRIPTION:Title: Monogenic fields with odd class number\nby Artane Siad (IAS/Prin
ceton) 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\, condit
ional on a widely expected tail estimate\, compute this average exactly. A
s an application\, we show that there are infinitely many number fields wi
th odd class number in any even degree and signature. Time permitting\, we
will also discuss extensions of this result to orders and to the relative
setting.\n
LOCATION:https://researchseminars.org/talk/number_theory/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ellen Eischen (U Oregon)
DTSTART:20211201T230000Z
DTEND:20211202T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/7
DESCRIPTION:Title: p-adic aspects of L-functions\, with a view toward Spin L-functions<
/a>\nby Ellen Eischen (U Oregon) as part of UBC Number theory seminar\n\n\
nAbstract\nThe study of p-adic properties of values of L-functions dates b
ack to Kummer's study of congruences between values of the Riemann zeta fu
nction at negative odd\nintegers\, as part of his attempt to understand cl
ass numbers of cyclotomic extensions.\nAfter Kummer's ideas largely lay do
rmant for over a half century\, Iwasawa's conjectures\nabout the meaning o
f p-adic L-functions led to renewed interest\, and Serre's discovery of\np
-adic modular forms opened up a new approach to studying congruences betwe
en values\nof L-functions\, forming the foundation for continued developme
nts today.\n\nWith a viewpoint that encompasses several settings\, includi
ng modular forms (on $GL_2$)\nand automorphic forms on higher rank groups\
, I will introduce p-adic L-functions and a\nrecipe for constructing them\
, which relies partly on properties of Fourier coefficients of modular (an
d automorphic) forms. Along the way\, I will introduce several recent deve
lopments\nand put them in the context of constructions of Serre\, Katz\, a
nd Hida. As an example of a\nrecent application of these ideas\, I will di
scuss the results of a paper-in-preparation\, joint\nwith G. Rosso and S.
Shah\, on p-adic Spin L-functions of ordinary cuspidal automorphic\nrepres
entations of $GSp_6$ associated to Siegel modular forms.\n
LOCATION:https://researchseminars.org/talk/number_theory/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rylan Gajek-Leonard (UMass Amherst)
DTSTART:20211006T220000Z
DTEND:20211006T230000Z
DTSTAMP:20260422T225924Z
UID:number_theory/8
DESCRIPTION:Title: Iwasawa Invariants of Modular Forms with $a_p=0$\nby Rylan Gajek
-Leonard (UMass Amherst) as part of UBC Number theory seminar\n\n\nAbstrac
t\nMazur-Tate elements provide a convenient method to study the\nanalytic
Iwasawa theory of p-nonordinary modular forms\, where the\nassociated p-ad
ic L-functions tend to have unbounded coefficients. The\nIwasawa invariant
s of Mazur-Tate elements are well-understood in the case\nof weight 2 modu
lar forms\, where they can be related to the growth of\np-Selmer groups an
d decompositions of the p-adic L-function. At higher\nweights\, less is kn
own. By constructing certain lifts to the full Iwasawa\nalgebra\, we compu
te the Iwasawa invariants of Mazur-Tate elements for\nhigher weight modula
r 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 relat
ion between the plus/minus invariants\nat weights 2 and p+1.\n
LOCATION:https://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:20260422T225924Z
UID:number_theory/9
DESCRIPTION:Title: The number of directions determined by a Cartesian product in the af
fine Galois plane\nby Ethan White & Chi Hoi Yip (University of British
Columbia) as part of UBC Number theory seminar\n\n\nAbstract\nThe directi
ons determined by a subset $U \\subset AG(2\,p)$ is the set of slopes\nfor
med by pairs of points from $U$. For $U = A \\times B$\, a Cartesian produ
ct\,\nwe give a new lower bound on the number of directions determined by
$U$.\nCombining this result with estimates on exponential sums\, we make p
rogress on\nthe Paley graph conjecture (a double character sum estimate).\
n\nWhen $A=B$ is an arithmetic progression\, we give an asymptotic formula
for the\nnumber of directions. Our method involves computing an asymptoti
c formula for\nthe number of solutions to the Diophantine equation $ad+bc
= p$.\n\nJoint work with Daniel Di Benedetto\, Greg Martin\, and Jozsef So
lymosi.\n\nSpeakers: Ethan White (UBC) homepage: https://personal.math.ubc
.ca/~epwhite/ \n\nChi Hoi Yip (UBC) homepage: https://sites.google.com/vie
w/kyle-chi-hoi-yip/home \n\nSince the site only allows for one homepage li
nk\, I went with that of the second named speaker (based on the last named
speaker given first priority principle).\n
LOCATION:https://researchseminars.org/talk/number_theory/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalia Garcia-Fritz (Pontificia Universidad Católica de Chile)
DTSTART:20211103T220000Z
DTEND:20211103T230000Z
DTSTAMP:20260422T225924Z
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 (Pontificia Un
iversidad Católica de Chile) as part of UBC Number theory seminar\n\n\nAb
stract\nAfter the solution by Davis\, Putnam\, Robinson and Matiyasevich o
f Hilbert's\nTenth problem for the integers\, a natural extension that rem
ains mostly open is the analogue for rings of integers of number fields. S
everal cases were proved in the seventies\nand eighties by Denef\, Lipshit
z\, Pheidas\, Shlapentokh\, Videla and Shapiro\, but after that\npoint the
re has been a long hiatus on unconditional results. Most recently\, ellipt
ic curve\ncriteria by Poonen\, Cornelissen-Pheidas-Zahidi and Shlapentokh
have led to a complete\nsolution under standard arithmetic conjectures\, t
hanks 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 recent\nt
echniques from Iwasawa theory and Heegner points.\n
LOCATION:https://researchseminars.org/talk/number_theory/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashay Burungale (Caltech/UT Austin)
DTSTART:20211124T231000Z
DTEND:20211125T001000Z
DTSTAMP:20260422T225924Z
UID:number_theory/11
DESCRIPTION:Title: A conjecture of Rubin\nby Ashay Burungale (Caltech/UT Austin) a
s part of UBC Number theory seminar\n\n\nAbstract\nIn 1987\, with an eye t
owards anticyclotomic Iwasawa theory of CM\nelliptic curves at inert prime
s\, Rubin proposed a basic conjecture on the\nstructure of Iwasawa module
of local units over anticyclotomic extensions\nof the unramified quadratic
extension of $\\mathbb{Q}_p$. The talk will report on\nrecent proof of Ru
bin's conjecture and some of subsequent developments\n(joint with S. Kobay
ashi and K. Ota).\n\nTo accommodate a special request\, we will be startin
g tomorrow's seminar at\n*3:10PM.*\n
LOCATION:https://researchseminars.org/talk/number_theory/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prajeet Bajpai (UBC)
DTSTART:20211208T230000Z
DTEND:20211209T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/12
DESCRIPTION:Title: Effective Methods for Norm-Form Equations\nby Prajeet Bajpai (U
BC) as part of UBC Number theory seminar\n\n\nAbstract\nLet $\\alpha_1\,\\
ldots\,\\alpha_k$ be linearly independent elements\nof a number field $K$
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\nbinary forms
was solved by Thue in 1909\, and the general case was resolved\nby Schmid
t in 1971 through his Subspace Theorem generalizing the work of\nThue-Sieg
el-Roth. Unfortunately these results are ineffective\, and do not\nprovide
any means of determining a bound on the height of exceptional\nsolutions-
- in particular\, they do not allow us to determine a complete\nlist of so
lutions for even a single norm-form equation.\n\nBaker's theorem on linear
forms in logarithms gave an effective version of\nThue's result for binar
y forms\, and Vojta in his PhD thesis was able extend\nthis effectivity to
three-variable norm-form equations under the assumption\nthat $K$ is tota
lly complex and Galois. In this talk we discuss effective\nresolution for
certain norm-form equations in four and five variables\,\nextending the wo
rk of Vojta. In particular\, we completely and effectively\nresolve the qu
estion of norm-form equations over totally complex Galois\nsextic fields.
The results are motivated by joint work with Mike Bennett.\n
LOCATION:https://researchseminars.org/talk/number_theory/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Hartmann (UPenn)
DTSTART:20211215T230000Z
DTEND:20211216T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/13
DESCRIPTION:Title: Local-global principles for linear algebraic groups over arithmetic
function fields\nby Julia Hartmann (UPenn) as part of UBC Number theo
ry seminar\n\n\nAbstract\nArithmetic function fields are one variable func
tion fields over complete\ndiscretely valued fields. They naturally admit
several collections of\noverfields with respect to which one can study loc
al-global principles. The\ntalk will concern such local-global principles
for torsors under linear\nalgebraic groups\, as well as their obstructions
. (Joint work with\nJ.L.-Colliot-Thélène\, D. Harbater\, D. Krashen\, R.
Parimala\, and V. Suresh.)\n
LOCATION:https://researchseminars.org/talk/number_theory/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbey Bourdon (Wake Forest University)
DTSTART:20211220T230000Z
DTEND:20211221T000000Z
DTSTAMP:20260422T225924Z
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 seminar\n\n\
nAbstract\nA common classification problem is to identify the groups which
arise as\nthe torsion subgroup of an elliptic curve defined over any numb
er field of\na fixed degree. That only finitely many such groups occur in
this context\nis a consequence of Merel's Uniform Boundedness Theorem. How
ever\, for\ncertain families of elliptic curves--such as those with comple
x\nmultiplication (CM)--recent advances have allowed us to move beyond a\n
fixed-degree classification to glimpse the behavior of torsion points over
\ninfinitely many degrees of a restricted form. In this talk\, I will disc
uss\nrecent work with Holly Paige Chaos which characterizes the groups tha
t\narise as torsion subgroups of CM elliptic curves defined over number fi
elds\nof degree 2p where p is prime. Here\, a classification in the strong
est\nsense is tied to determining whether there exist infinitely many Soph
ie\nGermain primes.\n
LOCATION:https://researchseminars.org/talk/number_theory/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Heidi Goodson (Brooklyn College)
DTSTART:20220112T230000Z
DTEND:20220113T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/15
DESCRIPTION:Title: Sato-Tate Groups in Dimension Greater than 3\nby Heidi Goodson
(Brooklyn College) as part of UBC Number theory seminar\n\n\nAbstract\nThe
focus of this talk is on Sato-Tate groups of abelian varieties --\ncompac
t groups predicted to determine the limiting distributions of local zeta\n
functions. In recent years\, complete classifications of Sato-Tate groups
in\ndimensions 1\, 2\, and 3 have been given\, but there are obstacles to
providing\nclassifications in higher dimensions. In this talk\, I will des
cribe my recent\nwork on families of higher dimensional Jacobian varieties
. This work is partly\njoint with Melissa Emory.\n
LOCATION:https://researchseminars.org/talk/number_theory/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State University)
DTSTART:20220119T230000Z
DTEND:20220120T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/16
DESCRIPTION:Title: Eisenstein congruences and class groups of pure fields\nby Pres
ton Wake (Michigan State University) as part of UBC Number theory seminar\
n\n\nAbstract\nLet $p$ and $N$ be primes and assume $N$ is $−1$ modulo $
p$. Then the class number\nof $\\mathbb{Q}(N^{1/p})$ is divisible by $p$.
I'll explain how to prove this using congruences between\nmodular forms. T
his is joint work with Jackie Lang\n
LOCATION:https://researchseminars.org/talk/number_theory/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Emory (University of Toronto)
DTSTART:20220126T230000Z
DTEND:20220127T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/17
DESCRIPTION:Title: Sato-Tate distributions for higher genus curves (TALK CANCELLED)\nby Melissa Emory (University of Toronto) as part of UBC Number theory s
eminar\n\n\nAbstract\nWe discuss work to determine Sato-Tate groups for hi
gher genus\ncurves. In so doing\, we detail an effective algorithm that c
omputes the\nidentity componenent of Sato-Tate groups.\nWe will also discu
ss open problems related to this work and graduate\nstudents are encourage
d to attend. This is joint work with H. Goodson and\nA.Peyrot.\n\nUnfortu
nately\, we have to cancel today's talk.\n
LOCATION:https://researchseminars.org/talk/number_theory/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matilde Lalin (Université de Montréal)
DTSTART:20220209T230000Z
DTEND:20220210T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/18
DESCRIPTION:Title: On the Northcott property of $L$-functions over function fields
\nby Matilde Lalin (Université de Montréal) as part of UBC Number theory
seminar\n\n\nAbstract\nThe Northcott property implies that a set of algeb
raic numbers with bounded height and bounded degree must be finite. Pazuki
and Pengo introduced a variant of the Northcott property for number field
s using special values of the Dedekind zeta function to measure the height
. We consider this question for global function fields with constant field
s $\\mathbb{F}_q$. This is joint work with Xavier Genereux and Wanlin Li.\
n
LOCATION:https://researchseminars.org/talk/number_theory/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly Isham (Colgate University)
DTSTART:20220202T230000Z
DTEND:20220203T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/19
DESCRIPTION:Title: Zeta functions and asymptotics related to subrings in $\\mathbb{Z}^
n$\nby Kelly Isham (Colgate University) as part of UBC Number theory s
eminar\n\n\nAbstract\nWe can define a zeta function of a group (or ring) t
o be the Dirichlet series associated to the sequence that counts the numbe
r of subgroups (or subrings) of a given index. The subgroup zeta function
over $\\mathbb{Z}^n$ is well-understood\, as is the asymptotic growth of s
ubgroups in $\\mathbb{Z}^n$. Much less is known about the subring zeta fun
ction over $\\mathbb{Z}^n$ and the asymptotic growth of subrings in $\\mat
hbb{Z}^n$. In this talk\, we discuss the progress toward answering this qu
estion and we give new lower bounds on the asymptotic growth of subrings i
n $\\mathbb{Z}^n$. We also define a similar zeta function corresponding to
subrings of corank at most k in $\\mathbb{Z}^n$. While the proportion of
subgroups in $\\mathbb{Z}^n$ of corank $k$ is positive for each $k$\, we s
how this is not the case for subrings in $\\mathbb{Z}^n$ of corank $k$ whe
n $n$ is sufficiently larger than $k$. Lastly\, we make connections to ord
ers in number fields. Part of this work is joint with Nathan Kaplan.\n
LOCATION:https://researchseminars.org/talk/number_theory/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ananth Shankar (University of Wisconsin Madison)
DTSTART:20220316T220000Z
DTEND:20220316T230000Z
DTSTAMP:20260422T225924Z
UID:number_theory/20
DESCRIPTION:Title: Canonical heights and the Andre-Oort conjecture\nby Ananth Shan
kar (University of Wisconsin Madison) as part of UBC Number theory seminar
\n\n\nAbstract\nLet S be a Shimura variety. The Andre-Oort conjecture posi
ts that the Zariski\nclosure of special points must be a sub-Shimura subva
riety of S. The Andre-Oort\nconjecture for $A_g$ (the moduli space of prin
cipally polarized Abelian\nvarieties) and therefore its sub-Shimura variet
ies was proved by Jacob\nTsimerman. However\, this conjecture was unknown
for Shimura varieties without a\nmoduli interpretation. I will describe jo
int work with Jonathan Pila and Jacob\nTsimerman (with an appendix by Esna
ult-Groechenig) where we prove the Andre-Oort conjecture in full generalit
y.\n
LOCATION:https://researchseminars.org/talk/number_theory/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cedric Dion (Univ. Laval)
DTSTART:20220309T230000Z
DTEND:20220310T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/21
DESCRIPTION:Title: Iwasawa theory for supersingular abelian varieties\nby Cedric D
ion (Univ. Laval) as part of UBC Number theory seminar\n\n\nAbstract\nFix
an odd prime number p. Let K be a quadratic imaginary field where p\nsplit
s. Let A be an abelian variety de\nned over K with good supersingular redu
ction at\nboth primes above p. In this talk\, we investigate some aspect o
f 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 rank of\
nA along this $\\mathbb{Z}_p^2$-extension grows like a function which is $
O(p^n\n)$ (joint with J. Ray). Finally\,\nusing the recently developed the
ory of gamma-systems\, we prove an algebraic functional\nequation involvin
g the Pontryagin dual of our Selmer groups.\n
LOCATION:https://researchseminars.org/talk/number_theory/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabella Negrini (Mcgill University)
DTSTART:20220216T230000Z
DTEND:20220217T000000Z
DTSTAMP:20260422T225924Z
UID:number_theory/22
DESCRIPTION:Title: A Shimura-Shintani correspondence for rigid analytic cocycles\n
by Isabella Negrini (Mcgill University) as part of UBC Number theory semin
ar\n\n\nAbstract\nIn their paper Singular moduli for real quadratic fields
: a rigid\n analytic approach\, Darmon and Vonk introduced rigid meromorph
ic 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 narrow rin
g class fields of real\n quadratic fiends and behave analogously to CM val
ues of modular functions\n on $SL_2(\\mathbb{Z})\\backslash H$. In this ta
lk I will present some progress towards\n developing a Shimura-Shintani co
rrespondence in this setting.\n
LOCATION:https://researchseminars.org/talk/number_theory/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liyang Yang (Princeton)
DTSTART:20220302T230000Z
DTEND:20220303T000000Z
DTSTAMP:20260422T225924Z
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 seminar\n\
n\nAbstract\nIn this talk\, we will introduce an explicit relative trace f
ormula to study central\nL-values on $U(3)×U(2)$. In conjunction with com
putations of local factors in Ichino-Ikeda\nformulas for Bessel periods\,
we obtain some important properties of these central L-values\nover certai
n family: the first moment\, nonvanishing and subconvexity in the level as
pect.\nThis is joint work with Philippe Michel and Dinakar Ramakrishnan.\n
LOCATION:https://researchseminars.org/talk/number_theory/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siddhi Pathak (Chennai Mathematical Institute)
DTSTART:20220324T173000Z
DTEND:20220324T183000Z
DTSTAMP:20260422T225924Z
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 theory semi
nar\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 this day.
In fact\, it is widely\nbelieved that the odd zeta-values do not satisfy
any polynomial relation with $\n\\pi $ over the rational numbers. Almost t
hree centuries after Euler\, several\ndifferent perspectives have emerged
to study the general case of special values\nof L-functions. In this talk\
, we discuss a more classical approach and describe\nrecent progress on re
lated conjectures by Chowla\, Erdos and Milnor.\n
LOCATION:https://researchseminars.org/talk/number_theory/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Heejong Lee (University of Toronto)
DTSTART:20220330T220000Z
DTEND:20220330T230000Z
DTSTAMP:20260422T225924Z
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 of UBC N
umber theory seminar\n\n\nAbstract\nThe conjectural mod p Langlands progra
m relates continuous mod p local Galois representations to smooth admissib
le representation of p-adic groups (with coeficient\nfield of characterist
ic p). Except for $GL_1$ and $GL_2(\\mathbb{Q}_p)$\, there is no known con
struction of\nmod p Langlands correspondence. However\, it is possible to
construct a candidate using the\nspace of mod p automorphic forms and the
Taylor-Wiles method. It is not clear whether\nthis candidate\, constructed
by taking non-canonical choices of global data\, is determined\nby the in
itial mod p local Galois representation.\nIn this talk\, we discuss a ques
tion in reverse direction: can we recover the initial\nlocal Galois repres
entation from the smooth admissible representation of p-adic group\nconstr
ucted above? We will discuss the proof of this statement in the case of $G
Sp_4(\\mathbb{Q}_p)$ and\nlocal Galois representations that are upper-tria
ngular\, maximally non-split\, and generic.\nOne of the main ingredients i
n the proof is explicit Jantzen filtration of lattices in a certain\ntame
type. This is joint work with John Enns.\n
LOCATION:https://researchseminars.org/talk/number_theory/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mishty Ray (University of Calgary)
DTSTART:20220406T220000Z
DTEND:20220406T230000Z
DTSTAMP:20260422T225924Z
UID:number_theory/26
DESCRIPTION:Title: Geometry of local Arthur packets for simple unramified Arthur param
eters for $GL_n$\nby Mishty Ray (University of Calgary) as part of UBC
Number theory seminar\n\n\nAbstract\nThe local Langlands correspondence f
or a connected reductive p-adic group G partitions the set of equivalence
classes of smooth irreducible representations of G(F) into L-packets using
equivalence classes of Langlands parameters. Vogan's geometric perspectiv
e gives us a moduli space of Langlands parameters\, and the correspondence
can be viewed as a relation between the set of equivalence classes of smo
oth irreducible representations of G(F) and simple objects in the category
of equivariant perverse sheaves on the moduli space of Langlands paramete
rs that share a common infinitesimal parameter. This geometry gives us the
notion of an ABV-packet\, a set of smooth irreducible representations of
G(F)\, which conjecturally generalizes the notion of a local Arthur packet
- a local Arthur packet is conjecturally an ABV-packet. In this talk\, we
will look at Langlands parameters coming from simple Arthur parameters in
the case of $GL_n$. We will explore the geometry of the moduli space of L
anglands parameters using an example. We will see work in progress towards
proving that the local Arthur packet is the ABV-packet for this case.\n
LOCATION:https://researchseminars.org/talk/number_theory/26/
END:VEVENT
END:VCALENDAR