BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Victoria Cantoral
DTSTART:20200727T163000Z
DTEND:20200727T170000Z
DTSTAMP:20260404T110911Z
UID:POINT/1
DESCRIPTION:Title: The Mumford—Tate conjecture implies the algebraic Sato—Tate conje
cture\nby Victoria Cantoral as part of POINT: New Developments in Numb
er Theory\n\n\nAbstract\nThe famous Mumford-Tate conjecture asserts that\,
for every prime number $\\ell$\, Hodge cycles are $\\mathbb{Q}_{\\ell}$-l
inear combinations of Tate cycles\, through Artin's comparisons theorems b
etween Betti and étale cohomology. The algebraic Sato-Tate conjecture\, i
ntroduced by Serre and developed later by Banaszak and Kedlaya\, is a powe
rful tool in order to prove new instances of the generalized Sato-Tate con
jecture. This previous conjecture is related with the equidistribution of
Frobenius traces.\n\nOur main goal is to prove that the Mumford-Tate conje
cture for an abelian variety A implies the algebraic Sato-Tate conjecture
for A. The relevance of this result lies mainly in the fact that the list
of known cases of the Mumford-Tate conjecture was up to now a lot longer t
han the list of known cases of the algebraic Sato-Tate conjecture. This is
a joint work with Johan Commelin.\n\nIf you like to attend the talk\, ple
ase register here: https://umich.zoom.us/meeting/register/tJAufuqtqDksG9fE
mjTbWHM4QOEUad6Ke-DE.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rahul Dalal
DTSTART:20200727T170000Z
DTEND:20200727T173000Z
DTSTAMP:20260404T110911Z
UID:POINT/2
DESCRIPTION:Title: Statistics of Automorphic Representations through Simplified Trace Fo
rmulas\nby Rahul Dalal as part of POINT: New Developments in Number Th
eory\n\n\nAbstract\nAutomorphic representations encode information about a
broad range of interesting mathematical objects. They are very difficult
to study individually so it is often good to study them in families instea
d. The Arthur-Selberg trace formula is a powerful tool for this. For certa
in very nice families (discrete series at infinity)\, the invariant and st
able versions of the trace formula take on a simpler form\, allowing us to
much more easily prove distributional results. I will discuss some of the
se results and the techniques used for the required trace formula computat
ions.\n\nIf you like to attend the talk\, please register here: https://um
ich.zoom.us/meeting/register/tJAufuqtqDksG9fEmjTbWHM4QOEUad6Ke-DE.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karol Koziol (University of Michigan)
DTSTART:20200824T130000Z
DTEND:20200824T133000Z
DTSTAMP:20260404T110911Z
UID:POINT/5
DESCRIPTION:Title: Supersingular representations of p-adic reductive groups.\nby Kar
ol Koziol (University of Michigan) as part of POINT: New Developments in N
umber Theory\n\n\nAbstract\nThe representation theory of p-adic reductive
groups plays an extremely important role in modern number theory. Namely\
, the local Langlands conjectures predict that (packets of) irreducible co
mplex representations of p-adic reductive groups (such as $\\mathrm{GL}_n(
\\mathbb{Q}_p)$\, $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)$\, etc.) should be pa
rametrized by certain representations of the Weil-Deligne group (a variant
of the usual absolute Galois group). A special role in this hypothetical
correspondence is held by the supercuspidal representations\, which gener
ically are expected to correspond to irreducible objects on the Galois sid
e\, and which serve as building blocks for all irreducible representations
. Motivated by recent advances in the mod-$p$ local Langlands program (i.
e.\, with mod-$p$ coefficients instead of complex coefficients)\, I will g
ive an overview of what is known about supersingular representations of $p
$-adic reductive groups\, which are the "mod-$p$ coefficients" analogs of
supercuspidal representations. This is joint work with Florian Herzig and
Marie-France Vigneras.\n\nPlease register for the talks on August 24 here
: \nhttps://virginia.zoom.us/meeting/register/tJMkc-uorT8iHdOXRaBkci8wHoKU
kqiXaq-E\n
LOCATION:https://stable.researchseminars.org/talk/POINT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Garen Chiloyan (University of Connecticut)
DTSTART:20200824T133000Z
DTEND:20200824T140000Z
DTSTAMP:20260404T110911Z
UID:POINT/6
DESCRIPTION:Title: A Classification of Isogeny-Torsion Graphs of Elliptic Curves Defined
over the Rationals\nby Garen Chiloyan (University of Connecticut) as
part of POINT: New Developments in Number Theory\n\n\nAbstract\nAn isogeny
graph is a nice visualization of the isogeny class of an elliptic curve.
A theorem of Kenku shows sharp bounds on the number of distinct isogenies
that a rational elliptic curve can have (in particular\, every isogeny gra
ph has at most 8 vertices). In this talk\, we give a complete classificati
on of the torsion subgroups over $\\mathbb{Q}$ that can occur in each vert
ex of a given isogeny graph of elliptic curves defined over the rationals.
This is joint work with \\'Alvaro Lozano-Robledo.\n\nPlease register for
the talks on August 24 here: \nhttps://virginia.zoom.us/meeting/register/t
JMkc-uorT8iHdOXRaBkci8wHoKUkqiXaq-E\n
LOCATION:https://stable.researchseminars.org/talk/POINT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:André Macedo (University of Reading)
DTSTART:20200909T000000Z
DTEND:20200909T003000Z
DTSTAMP:20260404T110911Z
UID:POINT/7
DESCRIPTION:Title: Local-global principles for norm equations\nby André Macedo (Uni
versity of Reading) as part of POINT: New Developments in Number Theory\n\
n\nAbstract\nGiven an extension L/K of number fields\, we say that the Has
se norm principle (HNP) holds if every non-zero element of K which is a no
rm everywhere locally is in fact a global norm from L. If L/K is cyclic\,
the original Hasse norm theorem states that the HNP holds. More generally\
, there is a cohomological description (due to Tate) of the obstruction to
the HNP for Galois extensions. In this talk\, I will present work develop
ing explicit methods to study this principle for non-Galois extensions as
well as some key applications in extensions whose normal closure has Galoi
s group A_n or S_n. I will additionally discuss the geometric interpretati
on of this concept and how it relates to the weak approximation property f
or norm varieties. If time permits\, I will also present some recent devel
opments on the statistics of the HNP\n
LOCATION:https://stable.researchseminars.org/talk/POINT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elisa Bellah (University of Oregon)
DTSTART:20200909T003000Z
DTEND:20200909T010000Z
DTSTAMP:20260404T110911Z
UID:POINT/8
DESCRIPTION:Title: Norm Form Equations and Linear Divisibility Sequences\nby Elisa B
ellah (University of Oregon) as part of POINT: New Developments in Number
Theory\n\n\nAbstract\nFinding integer solutions to norm form equations is
a classic Diophantine problem. Using the units of the associated coefficie
nt ring\, we can produce sequences of solutions to these equations. It tur
ns out that these solutions can be written as tuples of linear homogeneous
recurrence sequences\, each with characteristic polynomial equal to the m
inimal polynomial of our unit. We show that for certain families of norm f
orms\, these sequences are linear divisibility sequences.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jon Aycock (University of Oregon)
DTSTART:20200921T163000Z
DTEND:20200921T170000Z
DTSTAMP:20260404T110911Z
UID:POINT/9
DESCRIPTION:Title: Families of Differential Operators for Overconvergent Hilbert Modular
Forms\nby Jon Aycock (University of Oregon) as part of POINT: New Dev
elopments in Number Theory\n\n\nAbstract\nWe construct differential operat
ors for families of overconvergent Hilbert modular forms by interpolating
the Gauss--Manin connection on strict neighborhoods of the ordinary locus.
This is related to work done by Harron and Xiao and by Andreatta and Iovi
ta in the case of modular forms and has applications in particular to p-ad
ic L-functions of CM fields.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neelam Saikia (Indian Institute of Technology Guwahati)
DTSTART:20200921T170000Z
DTEND:20200921T173000Z
DTSTAMP:20260404T110911Z
UID:POINT/10
DESCRIPTION:Title: Zeros of $p$-adic hypergeometric series\nby Neelam Saikia (India
n Institute of Technology Guwahati) as part of POINT: New Developments in
Number Theory\n\n\nAbstract\nLet $p$ be an odd prime. McCarthy initiated a
study of hypergeometric functions in the $p$-adic setting. This function
can be understood as $p$-adic analogue of Gauss' hypergeometric function\,
and some kind of generalisation of Greene's hypergeometric function over
finite fields. In this talk we investigate arithmetic properties of certai
n families of McCarthy's hypergeometric functions. In particular\, we expl
icitly discuss all the possible values of these functions. Moreover\, we d
iscuss zeros of these functions.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seoyoung Kim (Queen's University)
DTSTART:20200810T163000Z
DTEND:20200810T170000Z
DTSTAMP:20260404T110911Z
UID:POINT/11
DESCRIPTION:Title: From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture<
/a>\nby Seoyoung Kim (Queen's University) as part of POINT: New Developmen
ts in Number Theory\n\n\nAbstract\nLet $E$ be an elliptic curve over $\\ma
thbb{Q}$ with discriminant\, and let $a_p$ be the Frobenius trace for each
prime p. In 1965\, Birch and Swinnerton-Dyer formulated a conjecture whic
h implies\n\n$\\lim\\limits_{x \\rightarrow \\infty} \\frac{1}{\\log x} \\
sum_{p< x} \\frac{a_p\\log p}{p}=-r+\\frac{1}{2}\,$\n\nwhere $r$ is the or
der of the zero of the $L$-function of $E$ at $s=1$\, which is predicted t
o be the Mordell-Weil rank of $E(\\mathbb{Q})$. We show that if the above
limit exits\, then the limit equals $-r+\\frac{1}{2}$\, and study the conn
ections to Riemann hypothesis for $E$. We also relate this to Nagao's conj
ecture. This is a recent joint work with M. Ram Murty.\n\nPlease register
for the talks on August 10 here:\nhttps://fordham.zoom.us/meeting/register
/tJwpde2srTgqHdYG6NMu5WmgzPiDnNJQMTsM\n
LOCATION:https://stable.researchseminars.org/talk/POINT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sachi Hashimoto (Boston University)
DTSTART:20200810T170000Z
DTEND:20200810T173000Z
DTSTAMP:20260404T110911Z
UID:POINT/12
DESCRIPTION:Title: Computing rational points on databases of curves\nby Sachi Hashi
moto (Boston University) as part of POINT: New Developments in Number Theo
ry\n\n\nAbstract\nFor a curve of genus at least $2$\, we know from Falting
s's theorem that its set of rational points is finite. A major challenge i
s to provably determine\, for a given curve\, this set of rational points.
One promising method is the Chabauty-Coleman method\, which uses $p$-adic
(Coleman) integrals to compute a finite set of p-adic points on the curve
including the rational points. We will discuss computations using the Cha
bauty-Coleman method to provably determine rational point sets for databas
es of certain genus $3$ superelliptic curves. This is joint work with Mari
a de Frutos Fernandez and Travis Morrison.\n\nPlease register for the talk
s on August 10 here:\nhttps://fordham.zoom.us/meeting/register/tJwpde2srTg
qHdYG6NMu5WmgzPiDnNJQMTsM\n
LOCATION:https://stable.researchseminars.org/talk/POINT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julie Desjardins (University of Toronto Mississauga)
DTSTART:20201005T130000Z
DTEND:20201005T133000Z
DTSTAMP:20260404T110911Z
UID:POINT/13
DESCRIPTION:Title: Density of rational points on a family of del Pezzo surface of degre
e 1\nby Julie Desjardins (University of Toronto Mississauga) as part o
f POINT: New Developments in Number Theory\n\n\nAbstract\nLet $k$ be a num
ber field and $X$ an algebraic variety over $k$. We want to study the set
of $k$-rational points $X(k)$. For example\, is $X(k)$ empty? If not\, is
it dense with respect to the Zariski topology? Del Pezzo surfaces are clas
sified by their degrees $d$ (an integer between 1 and 9). Manin and variou
s authors proved that for all del Pezzo surfaces of degree $d>1$\, $X(k)$
is dense provided that the surface has a $k$-rational point (that lies out
side a specific subset of the surface for $d=2$). For $d=1$\, the del Pezz
o surface always has a rational point. However\, we don't know if the set
of rational points is Zariski-dense. In this talk\, I present a result tha
t is joint with Rosa Winter in which we prove the density of rational poin
ts for a specific family of del Pezzo surfaces of degree 1 over $k$.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mingming Zhang (Oklahoma State University)
DTSTART:20201005T133000Z
DTEND:20201005T140000Z
DTSTAMP:20260404T110911Z
UID:POINT/14
DESCRIPTION:Title: Mahler Measure and its behavior under iteration\nby Mingming Zha
ng (Oklahoma State University) as part of POINT: New Developments in Numbe
r Theory\n\n\nAbstract\nFor an algebraic number $\\alpha$ we denote by $M(
\\alpha)$ the Mahler measure of $\\alpha$. As $M(\\alpha)$ is again an alg
ebraic number (indeed\, an algebraic integer)\, $M(\\cdot)$ is a self-map
on $\\overline{\\mathbb{Q}}$\, and therefore defines a dynamical system. T
he $\\mathit{orbit}$ $\\mathit{size}$ of $\\alpha$\, denoted $\\# \\mathca
l{O}_M(\\alpha)$\, is the cardinality of the forward orbit of $\\alpha$ un
der $M$. In this talk\, we will start by introducing the definition of Mah
ler measure\, briefly discuss results on the orbit sizes of algebraic num
bers with degree at least 3 and non-unit norm\, then we will turn our focu
s to the behavior of algebraic units\, which are of interest in Lehmer's p
roblem. We will mention the results regarding algebraic units of degree 4
and discuss that if $\\alpha$ is an algebraic unit of degree $d\\geq 5$ su
ch that the Galois group of the Galois closure of $\\mathbb{Q}(\\alpha)$ c
ontains $A_d$\, then the orbit size must be 1\, 2 or $\\infty$. Furthermor
e\, we will show that there exists units with orbit size larger than 2! Th
is is joint work with Paul Fili and Lucas Pottmeyer.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano López (University of Utah)
DTSTART:20201102T180000Z
DTEND:20201102T183000Z
DTSTAMP:20260404T110911Z
UID:POINT/17
DESCRIPTION:Title: Counting elliptic curves with prescribed torsion over imaginary quad
ratic fields\nby Allechar Serrano López (University of Utah) as part
of POINT: New Developments in Number Theory\n\n\nAbstract\nA generalizatio
n of Mazur's theorem\, proved by Kamienny\, states that there are 26 possi
bilities for the torsion subgroup of an elliptic curve over quadratic exte
nsions of the rational numbers. We prove that if $G$ is isomorphic to one
of these subgroups then the elliptic curves up to height $X$ whose torsion
is isomorphic to $G$ is on the order of $X^{\\frac{1}{d}}$ where $d>1$.\
n
LOCATION:https://stable.researchseminars.org/talk/POINT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Iyengar (King's College London)
DTSTART:20201102T173000Z
DTEND:20201102T180000Z
DTSTAMP:20260404T110911Z
UID:POINT/18
DESCRIPTION:Title: mod p spectral Hecke algebras\nby Ashwin Iyengar (King's College
London) as part of POINT: New Developments in Number Theory\n\n\nAbstract
\nIn this talk I will discuss work in progress (for $\\textnormal{GL}_2(\\
mathbb{Q}_p)$) on describing the mod $p$ derived Hecke algebra attached to
a Serre weight\, as well as the mod $p$ spectral Hecke algebra attached t
o the corresponding crystalline deformation ring. These objects should act
compatibly on the cohomology of arithmetic groups. I will describe these
Hecke algebras and their actions in more detail.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayreena Bakhtawar (La Trobe University)
DTSTART:20201117T090000Z
DTEND:20201117T093000Z
DTSTAMP:20260404T110911Z
UID:POINT/19
DESCRIPTION:Title: Contribution to uniform Diophantine approximation via continued frac
tions\nby Ayreena Bakhtawar (La Trobe University) as part of POINT: Ne
w Developments in Number Theory\n\n\nAbstract\nDiophantine approximation i
s a branch of number theory which is concerned with the question of how we
ll can an irrational number be approximated by a rational?\nOne of the maj
or ingredients to study problems in Diophantine approximation is continued
fraction expansion as they provide quick and efficient way for finding go
od rational approximations to irrational numbers.\nI will discuss the rela
tionship between Diophantine approximation and the theory of continued fra
ctions. And along the way I will talk about some measure theoretic results
including the landmark results of Dirichlet (1842)\, Khintchine (1924)\,
and Jarnik (1931) theorems to the questions in continued fractions. These
enable us to improve the classical results by using continued fractions.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katharina Müller (University Göttingen)
DTSTART:20201117T093000Z
DTEND:20201117T100000Z
DTSTAMP:20260404T110911Z
UID:POINT/20
DESCRIPTION:Title: The split prime $\\mathbb{Z}_p$-extension of imaginary quadratic fie
lds\nby Katharina Müller (University Göttingen) as part of POINT: Ne
w Developments in Number Theory\n\n\nAbstract\nLet $K$ be an imaginary qua
dratic field and $p$ a rational prime that splits into $p_1$ and $p_2$. Th
en there is a unique $\\mathbb{Z}_p$ extension that is only ramified at on
e of the primes above $p$. We will shift this extension by an abelian exte
nsion over $L/ K$ to $L_{\\infty}$. Let $M$ be the maximal $p$-abelian $p_
1$-ramified extension of $L_{\\infty}$. Generalizing work of Schneps we wi
ll show that $Gal(M/L_{\\infty})$ is a finitely generated $\\mathbb{Z}_p$-
module. If time allows we will also discuss the main conjecture for these
extensions. Part of this talk is joint work with Vlad Crisan.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Fox (University of Oregon)
DTSTART:20201021T000000Z
DTEND:20201021T003000Z
DTSTAMP:20260404T110911Z
UID:POINT/21
DESCRIPTION:Title: Supersingular loci in moduli spaces of abelian varieties\nby Mar
ia Fox (University of Oregon) as part of POINT: New Developments in Number
Theory\n\n\nAbstract\nGiven a moduli space of abelian varieties in charac
teristic $p$\, for example the reduction modulo $p$ of a modular curve\, i
t is natural to ask: what points in this moduli space parametrize supersin
gular abelian varieties? These points define the supersingular locus of th
e moduli space. In this talk\, we'll see several examples of moduli spaces
of abelian varieties\, and we'll discuss the geometry of their supersingu
lar loci.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Gleason (University of California\, Berkeley)
DTSTART:20201021T003000Z
DTEND:20201021T010000Z
DTSTAMP:20260404T110911Z
UID:POINT/22
DESCRIPTION:Title: On the geometric connected components of local Shimura varieties
\nby Ian Gleason (University of California\, Berkeley) as part of POINT: N
ew Developments in Number Theory\n\n\nAbstract\nThrough the recent introdu
ction of the theory of diamonds\, P. Scholze was able to define local vers
ions of Shimura varieties. These are rigid-analytic spaces that generalize
the generic fiber of a Rapoport-Zink space. It is widely expected that th
e cohomology of these interesting spaces realizes instances of the Langlan
ds correspondence. In this talk we describe the geometric connected compon
ents of these moduli spaces and relate it to local class field theory.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arpit Bansal (Jawaharlal Nehru University)
DTSTART:20210201T173000Z
DTEND:20210201T180000Z
DTSTAMP:20260404T110911Z
UID:POINT/23
DESCRIPTION:Title: Large sieve with square moduli for Z[i].\nby Arpit Bansal (Jawah
arlal Nehru University) as part of POINT: New Developments in Number Theor
y\n\n\nAbstract\nThe large sieve inequality is of fundamental importance i
n analytic number theory. Its theory started with Linnik’s investigation
of the least quadratic non-residue modulo primes on average. These days\,
there is a whole zoo of large sieve inequalities in all kind of contexts
(for number fields\, function fields\, automorphic forms\, etc.). The larg
e sieve with restricted sets of moduli $q \\in \\mathbb{Z}$\, in particula
r with square moduli\, were investigated by L. Zhao and S. Baier. Moreover
\, the large sieve with square moduli has found many applications\, in par
ticular\, in questions regarding elliptic curves. The large sieve for addi
tive characters was extended to number fields by Huxley. In my talk\, I wi
ll give a summary of the classical large sieve with square moduli and pres
ent new extenstions to number field $\\mathbb{Q}[i]$ which have recently b
een established in joint work with Stephan Baier.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liem Nguyen (Louisiana State University)
DTSTART:20210119T203000Z
DTEND:20210119T210000Z
DTSTAMP:20260404T110911Z
UID:POINT/24
DESCRIPTION:Title: On the Values and Spectrum of Weil Sum of Binomials\nby Liem Ngu
yen (Louisiana State University) as part of POINT: New Developments in Num
ber Theory\n\n\nAbstract\nThe Weil sum of an additive character $\\mu$ ove
r a finite field $F$ is defined to be $W_{F\,s}(a)=\\sum_{x \\in F} \\mu(x
^s-ax)$ where $s$ is an integer coprime to $|F^*|$. The Weil spectrum coun
ts distinct values of the Weil sum as $a$ runs through the invertible elem
ents in the finite field. Determining the values of these sums and the siz
e of its spectrum give answers to long-standing problems in cryptography\,
coding and information theory. In this talk\, we prove a special case of
the Vanishing Conjecture of Helleseth ($1971$) on the presence of zero in
the Weil spectrum. We then propose a new conjecture on when the Weil spect
rum contains at least five elements\, and prove it for a certain class of
Weil sum.\n\nTo join the talks on January 19th\, please register here: \n
\nhttps://ucsd.zoom.us/meeting/register/tJIuf-6uqjoiH9eVbtLN9y1G9l0qEBmbDR
mV\n
LOCATION:https://stable.researchseminars.org/talk/POINT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Edmund Karasiewicz (Ben-Gurion University)
DTSTART:20210201T180000Z
DTEND:20210201T183000Z
DTSTAMP:20260404T110911Z
UID:POINT/25
DESCRIPTION:Title: The Twisted Satake Transform and the Casselman-Shalika Formula\n
by Edmund Karasiewicz (Ben-Gurion University) as part of POINT: New Develo
pments in Number Theory\n\n\nAbstract\nThe Fourier coefficients of automor
phic forms are an important object of study due to their connection to $L$
-functions. In the adelic framework\, constructions of $L$-functions invol
ving Fourier coefficients (e.g. Langland-Shahidi and Rankin-Selberg method
s) naturally lead to spherical Whittaker functions on $p$-adic groups. Thu
s we would like to understand these spherical Whittaker functions to bette
r understand $L$-functions.\n\nCasselman-Shalika determined a formula for
the spherical Whittaker functions\, and basic algebraic manipulations reve
al that their formula can be more succinctly expressed in terms of charact
ers of the Langlands dual group. We will describe a new proof of the Casse
lman-Shalika formula that provides a conceptual explanation of the appeara
nce of characters.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Booher (University of Canterbury)
DTSTART:20210119T210000Z
DTEND:20210119T213000Z
DTSTAMP:20260404T110911Z
UID:POINT/26
DESCRIPTION:Title: Invariants in Towers of Curves over Finite Fields\nby Jeremy Boo
her (University of Canterbury) as part of POINT: New Developments in Numbe
r Theory\n\n\nAbstract\nA $Z_p$ tower of curves in characteristic $p$ is a
sequence $C_0\, C_1\, C_2\, ...$ of smooth projective curves over a perfe
ct field of characteristic $p$ such that $C_n$ is a branched cover of $C_{
n-1}$ and $C_n$ is a branched Galois $Z/(p^n)$-cover of $C_0$. For nice e
xamples of $Z_p$ towers\, the growth of the genus is stable: for sufficien
tly large $n$\, the genus of $C_n$ is a quadratic polynomial in $p^n$. In
characteristic $p$\, there are additional curve invariants like the a-num
ber which are poorly understood. They describe the group-scheme structure
of the $p$-torsion of the Jacobian. I will discuss work in progress with
Bryden Cais studying these invariants and suggesting that their growth is
also stable in genus stable $Z_p$ towers. This is a new kind of Iwasawa
theory for function fields.\n\nTo join the talks on January 19th\, please
register here: \n\nhttps://ucsd.zoom.us/meeting/register/tJIuf-6uqjoiH9eV
btLN9y1G9l0qEBmbDRmV\n
LOCATION:https://stable.researchseminars.org/talk/POINT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga (University of Cambridge)
DTSTART:20210215T130000Z
DTEND:20210215T133000Z
DTSTAMP:20260404T110911Z
UID:POINT/27
DESCRIPTION:Title: Selmer groups of some families of genus 3 curves and abelian surface
s\nby Jef Laga (University of Cambridge) as part of POINT: New Develop
ments in Number Theory\n\n\nAbstract\nManjul Bhargava and Arul Shankar hav
e determined the average size of the $n$-Selmer group of the family of all
elliptic curves over $\\mathbb{Q}$ ordered by height\, for $n$ at most $5
$. In this talk we will consider a family of nonhyperelliptic genus $3$ cu
rves\, and bound the average size of the $2$-Selmer group of their Jacobia
ns. This implies that a majority of curves in this family have relatively
few rational points. We also consider a family of abelian surfaces which a
re not principally polarized and obtain similar results. The proof is a co
mbination of the theory of simple singularities\, graded Lie algebras and
orbit-counting techniques.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Clifton (Emory University)
DTSTART:20210215T133000Z
DTEND:20210215T140000Z
DTSTAMP:20260404T110911Z
UID:POINT/28
DESCRIPTION:Title: An exponential bound for exponential diffsequences\nby Alexander
Clifton (Emory University) as part of POINT: New Developments in Number T
heory\n\n\nAbstract\nA theorem of van der Waerden states that for any posi
tive integer $r$\, if you partition $\\mathbf{N}$ into $r$ disjoint subset
s\, then one of them will contain arbitrarily long arithmetic progressions
. It is natural to ask what other arithmetic structures are preserved when
partitioning $\\mathbf{N}$ into a finite number of disjoint sets and to p
ose quantitative questions about these. We consider $D$-diffsequences\, in
troduced by Landman and Robertson\, which are increasing sequences in whic
h the consecutive differences all lie in some given set $D$. Here\, we con
sider the case where $D$ consists of all powers of $2$ and define $f(k)$ t
o be the smallest $n$ such that partitioning $\\{1\,2\,\\cdots\,n\\}$ into
$2$ subsets guarantees the presence of a $D$-diffsequence of length $k$ c
ontained entirely within one subset. We establish that $f(k)$ grows expone
ntially.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maryam Khaqan (Emory University)
DTSTART:20210303T003000Z
DTEND:20210303T010000Z
DTSTAMP:20260404T110911Z
UID:POINT/29
DESCRIPTION:Title: Elliptic Curves and Thompson's Sporadic Group\nby Maryam Khaqan
(Emory University) as part of POINT: New Developments in Number Theory\n\n
\nAbstract\nMoonshine began as a series of numerical coincidences connecti
ng finite groups to modular forms. It has since evolved into a rich theory
that sheds light on the underlying structures that these coincidences ref
lect.\n\n\nWe prove the existence of one such structure\, a module for the
Thompson group\, whose graded traces are specific half-integral weight we
akly holomorphic modular forms. We then proceed to use this module to stud
y the ranks of certain\nfamilies of elliptic curves. In particular\, this
serves as an example of moonshine being used to answer questions in number
theory.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Teppei Takamatsu (University of Tokyo)
DTSTART:20210303T010000Z
DTEND:20210303T013000Z
DTSTAMP:20260404T110911Z
UID:POINT/30
DESCRIPTION:Title: On finiteness of twisted forms of hyperkähler varieties\nby Tep
pei Takamatsu (University of Tokyo) as part of POINT: New Developments in
Number Theory\n\n\nAbstract\nFor a finite field extension $L/K$ and a vari
ety $X$ over $K$\,\nlet $Tw_{L/K} (X)$ be the set of isomorphism classes o
f varieties $Y$\nover $K$ which are isomorphic to $X$ after the base chang
e to $L$ (i.e.\nthe set of twisted forms of $X$ via $L/K$). In this talk\,
we prove the\nfiniteness of $Tw_{L/K}$ for K3 surfaces of characteristic
away from 2\nand hyperkähler varieties of characteristic 0. This work is
a\ngeneralization of Cattaneo-Fu's work on real forms of hyperkähler\nvar
ieties. We also give an application to the finiteness of derived\nequivale
nt twisted forms of hyperkähler varieties.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bella Tobin (Oklahoma State University)
DTSTART:20210601T200000Z
DTEND:20210601T203000Z
DTSTAMP:20260404T110911Z
UID:POINT/33
DESCRIPTION:Title: Reduction of post-critically finite polynomials\nby Bella Tobin
(Oklahoma State University) as part of POINT: New Developments in Number T
heory\n\n\nAbstract\nPost-critically finite maps are described as dynamica
l analogs of CM Abelian Varieties. A CM abelian varieties over a number fi
eld $K$ has everywhere good reduction in some finite extension $L/K$. This
motivates us to ask the question: do PCF maps have good reduction? We can
use a particular family of maps\, dynamical Belyi polynomials\, to provi
de necessary and sufficient conditions for a PCF polynomial of degree $d$
to have potential good reduction at a prime $p$. This is joint work with J
acqueline Anderson and Michelle Manes.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shaoyun Yi (University of South Carolina)
DTSTART:20210601T203000Z
DTEND:20210601T210000Z
DTSTAMP:20260404T110911Z
UID:POINT/34
DESCRIPTION:Title: On counting cuspidal automorphic representations of GSp(4)\nby S
haoyun Yi (University of South Carolina) as part of POINT: New Development
s in Number Theory\n\n\nAbstract\nThere are some well-known classical equi
distribution results like Sato-Tate conjecture for elliptic curves and equ
idistribution of Hecke eigenvalues of elliptic cusp forms. In this talk\,
we will discuss a similar equidistribution result for a family of cuspidal
automorphic representations for GSp(4). We formulate our theorem explicit
ly in terms of the number of cuspidal automorphic representations for GSp(
4) with certain conditions at the local places. To count the number of the
se cuspidal automorphic representations\, we will explore the connection b
etween Siegel cusp forms of degree 2 and cuspidal automorphic representati
ons of GSp(4). This is a joint work with Manami Roy and Ralf Schmidt.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Battistoni (Universite de Framche-Comte)
DTSTART:20210615T123000Z
DTEND:20210615T130000Z
DTSTAMP:20260404T110911Z
UID:POINT/35
DESCRIPTION:Title: On elliptic curves over $\\mathbb{Q}(T)$ and their ranks\nby Fra
ncesco Battistoni (Universite de Framche-Comte) as part of POINT: New Deve
lopments in Number Theory\n\n\nAbstract\nWe consider elliptic curves over
$\\mathbb{Q}(T)$ admitting Weierstrass model with coefficients being polyn
omials of small degree\, so that they are rational elliptic surfaces. In j
oint work with Sandro Bettin and Christophe Delaunay\, we apply Nagao's fo
rmula in order to detect the value of their ranks: this approach is orthog
onal to other geometric investigations\, and gives the values of the ranks
by looking at purely algebraic properties like the factorization of some
integer polynomials. We also prove that\, whenever restricting to some spe
cific families of curves\, the generic curve in these families has rank $0
$.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geoffrey Akers (CUNY Graduate Center)
DTSTART:20210615T130000Z
DTEND:20210615T133000Z
DTSTAMP:20260404T110911Z
UID:POINT/36
DESCRIPTION:Title: On a universal deformation ring that is a discrete valuation ring\nby Geoffrey Akers (CUNY Graduate Center) as part of POINT: New Developm
ents in Number Theory\n\n\nAbstract\nWe consider a crystalline universal d
eformation ring $R$ of an $n$-dimensional\, mod $p$ Galois representation
whose semisimplification is the direct sum of two non-isomorphic absolutel
y irreducible representations. Under some hypotheses\, we obtain that $R$
is a discrete valuation ring. The method examines the ideal of reducibilit
y of $R$\, which is used to construct extensions of representations in a S
elmer group with specified dimension. This can be used to deduce modulari
ty of representations.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ross Paterson (University of Glasgow)
DTSTART:20210628T163000Z
DTEND:20210628T170000Z
DTSTAMP:20260404T110911Z
UID:POINT/37
DESCRIPTION:Title: Average Ranks of Elliptic Curves after $p$-Extension\nby Ross Pa
terson (University of Glasgow) as part of POINT: New Developments in Numbe
r Theory\n\n\nAbstract\nAs $E$ varies among elliptic curves defined over t
he rational numbers\, a theorem of Bhargava and Shankar shows that the ave
rage rank of the Mordell--Weil group $E(\\mathbb{Q})$ is bounded. If we f
ix a number field $K$\, it is natural to then ask: is the average rank of
$E(K)$ also bounded in this family? Moreover\, how does the average rank
of $E(K)$ depend on $K$?\nThis talk will discuss recent progress on these
questions for a restricted set of $K$.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damián Gvirtz (University College London)
DTSTART:20210628T170000Z
DTEND:20210628T173000Z
DTSTAMP:20260404T110911Z
UID:POINT/38
DESCRIPTION:Title: The potential Hilbert Property for Enriques surfaces\nby Damián
Gvirtz (University College London) as part of POINT: New Developments in
Number Theory\n\n\nAbstract\nWhen does an algebraic variety have "many" ra
tional points? A possible formalisation of this notion is the (weak) Hilbe
rt Property for algebraic varieties\, a generalisation of Hilbert's classi
cal irreducibility theorem. I will report on joint work in progress with G
. Mezzedimi about a conjecture due to Campana and Corvaja-Zannier which co
ncerns this property in the case of Enriques surfaces.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tung Nguyen (Western University)
DTSTART:20220215T213000Z
DTEND:20220215T220000Z
DTSTAMP:20260404T110911Z
UID:POINT/39
DESCRIPTION:Title: Fekete polynomials\, quadratic residues\, and arithmetics\nby Tu
ng Nguyen (Western University) as part of POINT: New Developments in Numbe
r Theory\n\n\nAbstract\nFekete polynomials play an important role in the s
tudy of special values of L-functions of quadratic fields. While their ana
lytic properties are well-studied in the literature\, little is known abou
t their arithmetics. In this talk\, we will discuss some surprising arithm
etical properties of these polynomials. In particular\, we will see that s
pecial values of Fekete polynomials contain some rich information about th
e class numbers of quadratic fields. Furthermore\, their Galois groups se
em to follow a rather simple pattern. Time permitting\, I will discuss so
me recent progress on generalized Fekete polynomials. This is based on joi
nt work with Jan Minac and Nguyen Duy Tan.\n\nRegister for the next sessio
n of NDNT Round 5 on February 15 using the following links:\n\nhttps://uwm
adison.zoom.us/meeting/register/tJMkc-2hqDojG9yua5n-EkWxonnHyGeeMJkJ\n
LOCATION:https://stable.researchseminars.org/talk/POINT/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Upton (University of California San Diego)
DTSTART:20220215T220000Z
DTEND:20220215T223000Z
DTSTAMP:20260404T110911Z
UID:POINT/40
DESCRIPTION:Title: Newton Polygons of Artin-Schreier Coverings Curves\nby James Upt
on (University of California San Diego) as part of POINT: New Developments
in Number Theory\n\n\nAbstract\nLet $X$ be a smooth\, affine\, geometrica
lly connected curve over a finite field of characteristic $p > 2$. Let $C/
X$ be a finite Galois covering of degree p. A theorem of Kramer-Miller sta
tes that the p-adic Newton polygon NP($C$) is bounded below by a certain H
odge polygon HP($C$) which is defined in terms of local monodromy invarian
ts of $C/X$. Our main result is a local criterion that is necessary and su
fficient for NP($C$) and HP($C$) to coincide. Time permitting\, we will di
scuss some further results concerning the interaction of these two polygon
s. This is joint work with Joe Kramer-Miller.\n\nRegister for the next ses
sion of NDNT Round 5 on February 15 using the following links:\n\nhttps://
uwmadison.zoom.us/meeting/register/tJMkc-2hqDojG9yua5n-EkWxonnHyGeeMJkJ\n
LOCATION:https://stable.researchseminars.org/talk/POINT/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuki Yamamoto (The University of Tokyo)
DTSTART:20220228T113000Z
DTEND:20220228T120000Z
DTSTAMP:20260404T110911Z
UID:POINT/41
DESCRIPTION:Title: Comparing Bushnell-Kutzko and Sécherre's constructions of types for
$\\mathrm{GL}_{N}$ and its inner forms with Yu's construction\nby Yuk
i Yamamoto (The University of Tokyo) as part of POINT: New Developments in
Number Theory\n\n\nAbstract\nLet $F$ be a non-archimedean local field\, $
A$ be a central simple $F$-algebra\, and $G$ be the multiplicative group o
f $A$. To construct types for supercuspidal representations of $G$\, simp
le types by Sécherre and Yu's construction are already known. In this ta
lk\, we compare these constructions. In particular\, we show essentially
tame supercuspidal representations of $G$ defined by Bushnell-Henniart are
nothing but tame supercuspidal representations defined by Yu. This is a
joint work with Arnaud Mayeux.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lasse Grimmelt (University of Oxford)
DTSTART:20220228T120000Z
DTEND:20220228T123000Z
DTSTAMP:20260404T110911Z
UID:POINT/42
DESCRIPTION:Title: Goldbach's Problem with Almost Twin Primes\nby Lasse Grimmelt (U
niversity of Oxford) as part of POINT: New Developments in Number Theory\n
\n\nAbstract\nThe Binary Goldbach and Twin Prime conjectures have shaped t
he development of Analytic Number Theory in the the last century in fundam
ental manner. Among the strongest approximative results that we know are a
power saving bound on the exceptional set of the first conjecture\, prove
d by Montgomery and Vaughan\, and Chen’s result on almost Twin Primes. \
n\nIn this talk I present joint work with J. Teräväinen that aims to com
bine the two just mentioned results. More precisely\, we show a power savi
ng exceptional set bound for sums of two primes $p_1\, p_2$ such that $p_1
+2$ has at most 2\, $p_2+2$ at most 3 prime divisors. This improves previo
us results of this type in both strength of saving and number of prime div
isors of the shifted primes. Our proof uses a wide range of techniques. I
will give a sketch of how they play together and what the limitations are.
\n
LOCATION:https://stable.researchseminars.org/talk/POINT/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Arpin (University of Colorado Boulder)
DTSTART:20220321T150000Z
DTEND:20220321T153000Z
DTSTAMP:20260404T110911Z
UID:POINT/43
DESCRIPTION:Title: Adding Level Structure to Supersingular Elliptic Curve Isogeny Graph
s\nby Sarah Arpin (University of Colorado Boulder) as part of POINT: N
ew Developments in Number Theory\n\n\nAbstract\nSupersingular elliptic cur
ves have seen a resurgence in the past decade with new post-quantum crypto
graphic applications. In this talk\, we will discover why and how these cu
rves are used in new cryptographic protocol. Supersingular elliptic curve
isogeny graphs can be endowed with additional level structure. We will loo
k at the level structure graphs and the corresponding picture in a quatern
ion algebra.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weitong Wang (Harvard University)
DTSTART:20220321T153000Z
DTEND:20220321T160000Z
DTSTAMP:20260404T110911Z
UID:POINT/44
DESCRIPTION:Title: The bad part of the class group\nby Weitong Wang (Harvard Univer
sity) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nT
he Cohen-Lenstra-Martinet Heuristics provide us with basic ideas on the st
atistical results class groups of number fields. We can apply the heuristi
cs to the $p$-Sylow subgroups of the class groups when $p$ does not divide
the order of the Galois group. In this talk\, however\, I will first pres
ent an algebraic result on the structure of the $p$-Sylow subgroups of the
class groups when $p$ divides the Galois group. Then I will show that som
e of these primes can give us statistical results which show qualitative d
ifference from primes not dividing the order of the Galois group.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Kobin (Emory University)
DTSTART:20220404T213000Z
DTEND:20220404T220000Z
DTSTAMP:20260404T110911Z
UID:POINT/45
DESCRIPTION:Title: Zeta functions and decomposition spaces\nby Andrew Kobin (Emory
University) as part of POINT: New Developments in Number Theory\n\n\nAbstr
act\nZeta functions show up everywhere in math these days. While some work
in the past has brought homotopical methods into the theory of zeta funct
ions\, there is in fact a lesser-known zeta function that is native to hom
otopy theory. Namely\, every suitably finite decomposition space (aka 2-Se
gal space) admits an abstract zeta function as an element of its incidence
algebra. In this talk\, I will show how many 'classical' zeta functions f
rom number theory and algebraic geometry can be realized in this homotopic
al framework\, and briefly advertise some work in progress with Bogdan Krs
tic towards a motivic version of the above story.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tyler Billingsley (St. Olaf College)
DTSTART:20220404T220000Z
DTEND:20220404T223000Z
DTSTAMP:20260404T110911Z
UID:POINT/46
DESCRIPTION:Title: Recent Work on Specialization of Elliptic Surfaces\nby Tyler Bil
lingsley (St. Olaf College) as part of POINT: New Developments in Number T
heory\n\n\nAbstract\nThe study of cubic equations in two variables with at
least one rational solution\, i.e. the theory of elliptic curves\, is a c
entral area of study in modern number theory. The properties of specializa
tion of families of elliptic curves\, called elliptic surfaces\, is an are
a of current research\, in part because specialization was used by Elkies
to produce the current record for the highest known Mordell-Weil rank of a
n elliptic curve over $\\mathbb Q$. In this talk\, we will discuss a brief
history of and some recent developments in working effectively with speci
alization maps\, and in particular determining when they are injective.\n
LOCATION:https://stable.researchseminars.org/talk/POINT/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mar Curcó Iranzo (Utrecht University)
DTSTART:20220419T183000Z
DTEND:20220419T190000Z
DTSTAMP:20260404T110911Z
UID:POINT/47
DESCRIPTION:Title: Rational Torsion of Generalised Modular Jacobians\nby Mar Curcó
Iranzo (Utrecht University) as part of POINT: New Developments in Number
Theory\n\n\nAbstract\nWe consider the generalized Jacobian $J_0(N)_m$ of t
he modular curve $X_0(N)$ of level $N$\, with respect to the modulus $m$ c
onsisting of all cusps on the modular curve. When $N=p^(r)q^(s)$\, for $p$
and $q$ odd prime numbers\, we determine the group structure of the ratio
nal torsion of the Jacobian $J_0(N)_m$ up to 2-primary\, $p$-primary and $
q$-primary torsion. Our results extend known results for squarefree levels
and for prime power levels. Our proofs use their techniques\, as well as
results concerning the study of the rational points on the modular Jacobia
n and of the rational divisor class group of $X_0(N)$.\n\nRegister here: h
ttps://umanitoba.zoom.us/meeting/register/u5csf-CtqzstEtfTaoLl6L8gnaVIJnGV
B49w\n
LOCATION:https://stable.researchseminars.org/talk/POINT/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kelly Emmrich (Colorado State University)
DTSTART:20220419T190000Z
DTEND:20220419T193000Z
DTSTAMP:20260404T110911Z
UID:POINT/48
DESCRIPTION:Title: Non-splitting of the Hilbert exact sequence via a principal version
of the Chebotarev density theorem\nby Kelly Emmrich (Colorado State Un
iversity) as part of POINT: New Developments in Number Theory\n\n\nAbstrac
t\nLet $K/k$ be a Galois extension of number fields and $C$ a conjugacy cl
ass of $\\text{Gal}(K/k)$. In this talk\, we will investigate the density
of prime ideals of $k$ which factor as the product of principal ideals in
$K$ and have their associated Frobenius class equal to $C$. From this dens
ity we will determine a method for verifying the nonsplitting of the Hilbe
rt exact sequence.\n\nRegister here: https://umanitoba.zoom.us/meeting/reg
ister/u5csf-CtqzstEtfTaoLl6L8gnaVIJnGVB49w\n
LOCATION:https://stable.researchseminars.org/talk/POINT/48/
END:VEVENT
END:VCALENDAR