BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Victoria Cantoral
DTSTART;VALUE=DATE-TIME:20200727T163000Z
DTEND;VALUE=DATE-TIME:20200727T170000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/1
DESCRIPTION:Title: Th
e Mumford—Tate conjecture implies the algebraic Sato—Tate conjecture\nby Victoria Cantoral as part of POINT: New Developments in Number Theo
ry\n\n\nAbstract\nThe famous Mumford-Tate conjecture asserts that\, for ev
ery prime number $\\ell$\, Hodge cycles are $\\mathbb{Q}_{\\ell}$-linear c
ombinations of Tate cycles\, through Artin's comparisons theorems between
Betti and étale cohomology. The algebraic Sato-Tate conjecture\, introduc
ed by Serre and developed later by Banaszak and Kedlaya\, is a powerful to
ol in order to prove new instances of the generalized Sato-Tate conjecture
. This previous conjecture is related with the equidistribution of Frobeni
us traces.\n\nOur main goal is to prove that the Mumford-Tate conjecture f
or 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 know
n cases of the Mumford-Tate conjecture was up to now a lot longer than the
list of known cases of the algebraic Sato-Tate conjecture. This is a join
t work with Johan Commelin.\n\nIf you like to attend the talk\, please reg
ister here: https://umich.zoom.us/meeting/register/tJAufuqtqDksG9fEmjTbWHM
4QOEUad6Ke-DE.\n
LOCATION:https://researchseminars.org/talk/POINT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rahul Dalal
DTSTART;VALUE=DATE-TIME:20200727T170000Z
DTEND;VALUE=DATE-TIME:20200727T173000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/2
DESCRIPTION:Title: St
atistics of Automorphic Representations through Simplified Trace Formulas<
/a>\nby Rahul Dalal as part of POINT: New Developments in Number Theory\n\
n\nAbstract\nAutomorphic representations encode information about a broad
range of interesting mathematical objects. They are very difficult to stud
y individually so it is often good to study them in families instead. The
Arthur-Selberg trace formula is a powerful tool for this. For certain very
nice families (discrete series at infinity)\, the invariant and stable ve
rsions of the trace formula take on a simpler form\, allowing us to much m
ore easily prove distributional results. I will discuss some of these resu
lts and the techniques used for the required trace formula computations.\n
\nIf you like to attend the talk\, please register here: https://umich.zoo
m.us/meeting/register/tJAufuqtqDksG9fEmjTbWHM4QOEUad6Ke-DE.\n
LOCATION:https://researchseminars.org/talk/POINT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karol Koziol (University of Michigan)
DTSTART;VALUE=DATE-TIME:20200824T130000Z
DTEND;VALUE=DATE-TIME:20200824T133000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/5
DESCRIPTION:Title: Su
persingular representations of p-adic reductive groups.\nby Karol Kozi
ol (University of Michigan) as part of POINT: New Developments in Number T
heory\n\n\nAbstract\nThe representation theory of p-adic reductive groups
plays an extremely important role in modern number theory. Namely\, the l
ocal Langlands conjectures predict that (packets of) irreducible complex r
epresentations of p-adic reductive groups (such as $\\mathrm{GL}_n(\\mathb
b{Q}_p)$\, $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)$\, etc.) should be parametri
zed by certain representations of the Weil-Deligne group (a variant of the
usual absolute Galois group). A special role in this hypothetical corres
pondence is held by the supercuspidal representations\, which generically
are expected to correspond to irreducible objects on the Galois side\, and
which serve as building blocks for all irreducible representations. Moti
vated by recent advances in the mod-$p$ local Langlands program (i.e.\, wi
th mod-$p$ coefficients instead of complex coefficients)\, I will give an
overview of what is known about supersingular representations of $p$-adic
reductive groups\, which are the "mod-$p$ coefficients" analogs of supercu
spidal representations. This is joint work with Florian Herzig and Marie-
France Vigneras.\n\nPlease register for the talks on August 24 here: \nhtt
ps://virginia.zoom.us/meeting/register/tJMkc-uorT8iHdOXRaBkci8wHoKUkqiXaq-
E\n
LOCATION:https://researchseminars.org/talk/POINT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Garen Chiloyan (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20200824T133000Z
DTEND;VALUE=DATE-TIME:20200824T140000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/6
DESCRIPTION:Title: A
Classification of Isogeny-Torsion Graphs of Elliptic Curves Defined over t
he 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 theor
em of Kenku shows sharp bounds on the number of distinct isogenies that a
rational elliptic curve can have (in particular\, every isogeny graph has
at most 8 vertices). In this talk\, we give a complete classification of t
he torsion subgroups over $\\mathbb{Q}$ that can occur in each vertex of a
given isogeny graph of elliptic curves defined over the rationals. This i
s joint work with \\'Alvaro Lozano-Robledo.\n\nPlease register for the tal
ks on August 24 here: \nhttps://virginia.zoom.us/meeting/register/tJMkc-uo
rT8iHdOXRaBkci8wHoKUkqiXaq-E\n
LOCATION:https://researchseminars.org/talk/POINT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:André Macedo (University of Reading)
DTSTART;VALUE=DATE-TIME:20200909T000000Z
DTEND;VALUE=DATE-TIME:20200909T003000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/7
DESCRIPTION:Title: Lo
cal-global principles for norm equations\nby André Macedo (University
of Reading) as part of POINT: New Developments in Number Theory\n\n\nAbst
ract\nGiven an extension L/K of number fields\, we say that the Hasse norm
principle (HNP) holds if every non-zero element of K which is a norm ever
ywhere locally is in fact a global norm from L. If L/K is cyclic\, the ori
ginal Hasse norm theorem states that the HNP holds. More generally\, there
is a cohomological description (due to Tate) of the obstruction to the HN
P for Galois extensions. In this talk\, I will present work developing exp
licit methods to study this principle for non-Galois extensions as well as
some key applications in extensions whose normal closure has Galois group
A_n or S_n. I will additionally discuss the geometric interpretation of t
his concept and how it relates to the weak approximation property for norm
varieties. If time permits\, I will also present some recent developments
on the statistics of the HNP\n
LOCATION:https://researchseminars.org/talk/POINT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elisa Bellah (University of Oregon)
DTSTART;VALUE=DATE-TIME:20200909T003000Z
DTEND;VALUE=DATE-TIME:20200909T010000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/8
DESCRIPTION:Title: No
rm Form Equations and Linear Divisibility Sequences\nby Elisa Bellah (
University of Oregon) as part of POINT: New Developments in Number Theory\
n\n\nAbstract\nFinding integer solutions to norm form equations is a class
ic Diophantine problem. Using the units of the associated coefficient ring
\, we can produce sequences of solutions to these equations. It turns out
that these solutions can be written as tuples of linear homogeneous recurr
ence sequences\, each with characteristic polynomial equal to the minimal
polynomial of our unit. We show that for certain families of norm forms\,
these sequences are linear divisibility sequences.\n
LOCATION:https://researchseminars.org/talk/POINT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jon Aycock (University of Oregon)
DTSTART;VALUE=DATE-TIME:20200921T163000Z
DTEND;VALUE=DATE-TIME:20200921T170000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/9
DESCRIPTION:Title: Fa
milies of Differential Operators for Overconvergent Hilbert Modular Forms<
/a>\nby Jon Aycock (University of Oregon) as part of POINT: New Developmen
ts in Number Theory\n\n\nAbstract\nWe construct differential operators for
families of overconvergent Hilbert modular forms by interpolating the Gau
ss--Manin connection on strict neighborhoods of the ordinary locus. This i
s related to work done by Harron and Xiao and by Andreatta and Iovita in t
he case of modular forms and has applications in particular to p-adic L-fu
nctions of CM fields.\n
LOCATION:https://researchseminars.org/talk/POINT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neelam Saikia (Indian Institute of Technology Guwahati)
DTSTART;VALUE=DATE-TIME:20200921T170000Z
DTEND;VALUE=DATE-TIME:20200921T173000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/10
DESCRIPTION:Title: Z
eros of $p$-adic hypergeometric series\nby Neelam Saikia (Indian Insti
tute 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 so
me kind of generalisation of Greene's hypergeometric function over finite
fields. In this talk we investigate arithmetic properties of certain famil
ies of McCarthy's hypergeometric functions. In particular\, we explicitly
discuss all the possible values of these functions. Moreover\, we discuss
zeros of these functions.\n
LOCATION:https://researchseminars.org/talk/POINT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seoyoung Kim (Queen's University)
DTSTART;VALUE=DATE-TIME:20200810T163000Z
DTEND;VALUE=DATE-TIME:20200810T170000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/11
DESCRIPTION:Title: F
rom the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture\nby
Seoyoung Kim (Queen's University) as part of POINT: New Developments in N
umber Theory\n\n\nAbstract\nLet $E$ be an elliptic curve over $\\mathbb{Q}
$ with discriminant\, and let $a_p$ be the Frobenius trace for each prime
p. In 1965\, Birch and Swinnerton-Dyer formulated a conjecture which impli
es\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 order of
the zero of the $L$-function of $E$ at $s=1$\, which is predicted to be th
e Mordell-Weil rank of $E(\\mathbb{Q})$. We show that if the above limit e
xits\, then the limit equals $-r+\\frac{1}{2}$\, and study the connections
to Riemann hypothesis for $E$. We also relate this to Nagao's conjecture.
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/tJwpde
2srTgqHdYG6NMu5WmgzPiDnNJQMTsM\n
LOCATION:https://researchseminars.org/talk/POINT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sachi Hashimoto (Boston University)
DTSTART;VALUE=DATE-TIME:20200810T170000Z
DTEND;VALUE=DATE-TIME:20200810T173000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/12
DESCRIPTION:Title: C
omputing rational points on databases of curves\nby Sachi Hashimoto (B
oston University) as part of POINT: New Developments in Number Theory\n\n\
nAbstract\nFor a curve of genus at least $2$\, we know from Faltings's the
orem that its set of rational points is finite. A major challenge is to pr
ovably determine\, for a given curve\, this set of rational points. One pr
omising method is the Chabauty-Coleman method\, which uses $p$-adic (Colem
an) integrals to compute a finite set of p-adic points on the curve includ
ing the rational points. We will discuss computations using the Chabauty-C
oleman method to provably determine rational point sets for databases of c
ertain genus $3$ superelliptic curves. This is joint work with Maria de Fr
utos Fernandez and Travis Morrison.\n\nPlease register for the talks on Au
gust 10 here:\nhttps://fordham.zoom.us/meeting/register/tJwpde2srTgqHdYG6N
Mu5WmgzPiDnNJQMTsM\n
LOCATION:https://researchseminars.org/talk/POINT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julie Desjardins (University of Toronto Mississauga)
DTSTART;VALUE=DATE-TIME:20201005T130000Z
DTEND;VALUE=DATE-TIME:20201005T133000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/13
DESCRIPTION:Title: D
ensity of rational points on a family of del Pezzo surface of degree 1
\nby Julie Desjardins (University of Toronto Mississauga) as part of POINT
: New Developments in Number Theory\n\n\nAbstract\nLet $k$ be a number fie
ld 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 dens
e with respect to the Zariski topology? Del Pezzo surfaces are classified
by their degrees $d$ (an integer between 1 and 9). Manin and various autho
rs proved that for all del Pezzo surfaces of degree $d>1$\, $X(k)$ is dens
e provided that the surface has a $k$-rational point (that lies outside a
specific subset of the surface for $d=2$). For $d=1$\, the del Pezzo surfa
ce always has a rational point. However\, we don't know if the set of rati
onal points is Zariski-dense. In this talk\, I present a result that is jo
int with Rosa Winter in which we prove the density of rational points for
a specific family of del Pezzo surfaces of degree 1 over $k$.\n
LOCATION:https://researchseminars.org/talk/POINT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mingming Zhang (Oklahoma State University)
DTSTART;VALUE=DATE-TIME:20201005T133000Z
DTEND;VALUE=DATE-TIME:20201005T140000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/14
DESCRIPTION:Title: M
ahler Measure and its behavior under iteration\nby Mingming Zhang (Okl
ahoma State University) as part of POINT: New Developments in Number Theor
y\n\n\nAbstract\nFor an algebraic number $\\alpha$ we denote by $M(\\alpha
)$ the Mahler measure of $\\alpha$. As $M(\\alpha)$ is again an algebraic
number (indeed\, an algebraic integer)\, $M(\\cdot)$ is a self-map on $\\o
verline{\\mathbb{Q}}$\, and therefore defines a dynamical system. The $\\m
athit{orbit}$ $\\mathit{size}$ of $\\alpha$\, denoted $\\# \\mathcal{O}_M(
\\alpha)$\, is the cardinality of the forward orbit of $\\alpha$ under $M$
. In this talk\, we will start by introducing the definition of Mahler mea
sure\, briefly discuss results on the orbit sizes of algebraic numbers wi
th degree at least 3 and non-unit norm\, then we will turn our focus to th
e behavior of algebraic units\, which are of interest in Lehmer's problem.
We will mention the results regarding algebraic units of degree 4 and dis
cuss that if $\\alpha$ is an algebraic unit of degree $d\\geq 5$ such that
the Galois group of the Galois closure of $\\mathbb{Q}(\\alpha)$ contains
$A_d$\, then the orbit size must be 1\, 2 or $\\infty$. Furthermore\, we
will show that there exists units with orbit size larger than 2! This is j
oint work with Paul Fili and Lucas Pottmeyer.\n
LOCATION:https://researchseminars.org/talk/POINT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allechar Serrano López (University of Utah)
DTSTART;VALUE=DATE-TIME:20201102T180000Z
DTEND;VALUE=DATE-TIME:20201102T183000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/17
DESCRIPTION:Title: C
ounting elliptic curves with prescribed torsion over imaginary quadratic f
ields\nby Allechar Serrano López (University of Utah) as part of POIN
T: New Developments in Number Theory\n\n\nAbstract\nA generalization of Ma
zur's theorem\, proved by Kamienny\, states that there are 26 possibilitie
s for the torsion subgroup of an elliptic curve over quadratic extensions
of the rational numbers. We prove that if $G$ is isomorphic to one of thes
e subgroups then the elliptic curves up to height $X$ whose torsion is iso
morphic to $G$ is on the order of $X^{\\frac{1}{d}}$ where $d>1$.\n
LOCATION:https://researchseminars.org/talk/POINT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Iyengar (King's College London)
DTSTART;VALUE=DATE-TIME:20201102T173000Z
DTEND;VALUE=DATE-TIME:20201102T180000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/18
DESCRIPTION:Title: m
od p spectral Hecke algebras\nby Ashwin Iyengar (King's College London
) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nIn th
is 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 Serr
e weight\, as well as the mod $p$ spectral Hecke algebra attached to the c
orresponding crystalline deformation ring. These objects should act compat
ibly on the cohomology of arithmetic groups. I will describe these Hecke a
lgebras and their actions in more detail.\n
LOCATION:https://researchseminars.org/talk/POINT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayreena Bakhtawar (La Trobe University)
DTSTART;VALUE=DATE-TIME:20201117T090000Z
DTEND;VALUE=DATE-TIME:20201117T093000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/19
DESCRIPTION:Title: C
ontribution to uniform Diophantine approximation via continued fractions\nby Ayreena Bakhtawar (La Trobe University) as part of POINT: New Devel
opments in Number Theory\n\n\nAbstract\nDiophantine approximation is a bra
nch of number theory which is concerned with the question of how well can
an irrational number be approximated by a rational?\nOne of the major ingr
edients to study problems in Diophantine approximation is continued fracti
on expansion as they provide quick and efficient way for finding good rati
onal approximations to irrational numbers.\nI will discuss the relationshi
p between Diophantine approximation and the theory of continued fractions.
And along the way I will talk about some measure theoretic results includ
ing the landmark results of Dirichlet (1842)\, Khintchine (1924)\, and Jar
nik (1931) theorems to the questions in continued fractions. These enable
us to improve the classical results by using continued fractions.\n
LOCATION:https://researchseminars.org/talk/POINT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katharina Müller (University Göttingen)
DTSTART;VALUE=DATE-TIME:20201117T093000Z
DTEND;VALUE=DATE-TIME:20201117T100000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/20
DESCRIPTION:Title: T
he split prime $\\mathbb{Z}_p$-extension of imaginary quadratic fields
\nby Katharina Müller (University Göttingen) as part of POINT: New Devel
opments in Number Theory\n\n\nAbstract\nLet $K$ be an imaginary quadratic
field and $p$ a rational prime that splits into $p_1$ and $p_2$. Then ther
e is a unique $\\mathbb{Z}_p$ extension that is only ramified at one of th
e primes above $p$. We will shift this extension by an abelian extension o
ver $L/ K$ to $L_{\\infty}$. Let $M$ be the maximal $p$-abelian $p_1$-rami
fied extension of $L_{\\infty}$. Generalizing work of Schneps we will 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 extensi
ons. Part of this talk is joint work with Vlad Crisan.\n
LOCATION:https://researchseminars.org/talk/POINT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Fox (University of Oregon)
DTSTART;VALUE=DATE-TIME:20201021T000000Z
DTEND;VALUE=DATE-TIME:20201021T003000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/21
DESCRIPTION:Title: S
upersingular loci in moduli spaces of abelian varieties\nby Maria Fox
(University of Oregon) as part of POINT: New Developments in Number Theory
\n\n\nAbstract\nGiven a moduli space of abelian varieties in characteristi
c $p$\, for example the reduction modulo $p$ of a modular curve\, it is na
tural to ask: what points in this moduli space parametrize supersingular a
belian varieties? These points define the supersingular locus of the modul
i space. In this talk\, we'll see several examples of moduli spaces of abe
lian varieties\, and we'll discuss the geometry of their supersingular loc
i.\n
LOCATION:https://researchseminars.org/talk/POINT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Gleason (University of California\, Berkeley)
DTSTART;VALUE=DATE-TIME:20201021T003000Z
DTEND;VALUE=DATE-TIME:20201021T010000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/22
DESCRIPTION:Title: O
n the geometric connected components of local Shimura varieties\nby Ia
n Gleason (University of California\, Berkeley) as part of POINT: New Deve
lopments in Number Theory\n\n\nAbstract\nThrough the recent introduction o
f the theory of diamonds\, P. Scholze was able to define local versions of
Shimura varieties. These are rigid-analytic spaces that generalize the ge
neric fiber of a Rapoport-Zink space. It is widely expected that the cohom
ology of these interesting spaces realizes instances of the Langlands corr
espondence. In this talk we describe the geometric connected components of
these moduli spaces and relate it to local class field theory.\n
LOCATION:https://researchseminars.org/talk/POINT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arpit Bansal (Jawaharlal Nehru University)
DTSTART;VALUE=DATE-TIME:20210201T173000Z
DTEND;VALUE=DATE-TIME:20210201T180000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/23
DESCRIPTION:Title: L
arge sieve with square moduli for Z[i].\nby Arpit Bansal (Jawaharlal N
ehru University) as part of POINT: New Developments in Number Theory\n\n\n
Abstract\nThe large sieve inequality is of fundamental importance in analy
tic 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 nu
mber fields\, function fields\, automorphic forms\, etc.). The large sieve
with restricted sets of moduli $q \\in \\mathbb{Z}$\, in particular with
square moduli\, were investigated by L. Zhao and S. Baier. Moreover\, the
large sieve with square moduli has found many applications\, in particular
\, in questions regarding elliptic curves. The large sieve for additive ch
aracters was extended to number fields by Huxley. In my talk\, I will give
a summary of the classical large sieve with square moduli and present new
extenstions to number field $\\mathbb{Q}[i]$ which have recently been est
ablished in joint work with Stephan Baier.\n
LOCATION:https://researchseminars.org/talk/POINT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liem Nguyen (Louisiana State University)
DTSTART;VALUE=DATE-TIME:20210119T203000Z
DTEND;VALUE=DATE-TIME:20210119T210000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/24
DESCRIPTION:Title: O
n the Values and Spectrum of Weil Sum of Binomials\nby Liem Nguyen (Lo
uisiana State University) as part of POINT: New Developments in Number The
ory\n\n\nAbstract\nThe Weil sum of an additive character $\\mu$ over a fin
ite 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 counts dist
inct values of the Weil sum as $a$ runs through the invertible elements in
the finite field. Determining the values of these sums and the size of it
s spectrum give answers to long-standing problems in cryptography\, coding
and information theory. In this talk\, we prove a special case of the Van
ishing Conjecture of Helleseth ($1971$) on the presence of zero in the Wei
l spectrum. We then propose a new conjecture on when the Weil spectrum con
tains at least five elements\, and prove it for a certain class of Weil su
m.\n\nTo join the talks on January 19th\, please register here: \n\nhttps
://ucsd.zoom.us/meeting/register/tJIuf-6uqjoiH9eVbtLN9y1G9l0qEBmbDRmV\n
LOCATION:https://researchseminars.org/talk/POINT/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Edmund Karasiewicz (Ben-Gurion University)
DTSTART;VALUE=DATE-TIME:20210201T180000Z
DTEND;VALUE=DATE-TIME:20210201T183000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/25
DESCRIPTION:Title: T
he Twisted Satake Transform and the Casselman-Shalika Formula\nby Edmu
nd Karasiewicz (Ben-Gurion University) as part of POINT: New Developments
in Number Theory\n\n\nAbstract\nThe Fourier coefficients of automorphic fo
rms are an important object of study due to their connection to $L$-functi
ons. In the adelic framework\, constructions of $L$-functions involving Fo
urier coefficients (e.g. Langland-Shahidi and Rankin-Selberg methods) natu
rally lead to spherical Whittaker functions on $p$-adic groups. Thus we wo
uld like to understand these spherical Whittaker functions to better under
stand $L$-functions.\n\nCasselman-Shalika determined a formula for the sph
erical Whittaker functions\, and basic algebraic manipulations reveal that
their formula can be more succinctly expressed in terms of characters of
the Langlands dual group. We will describe a new proof of the Casselman-Sh
alika formula that provides a conceptual explanation of the appearance of
characters.\n
LOCATION:https://researchseminars.org/talk/POINT/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Booher (University of Canterbury)
DTSTART;VALUE=DATE-TIME:20210119T210000Z
DTEND;VALUE=DATE-TIME:20210119T213000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/26
DESCRIPTION:Title: I
nvariants in Towers of Curves over Finite Fields\nby Jeremy Booher (Un
iversity of Canterbury) as part of POINT: New Developments in Number Theor
y\n\n\nAbstract\nA $Z_p$ tower of curves in characteristic $p$ is a sequen
ce $C_0\, C_1\, C_2\, ...$ of smooth projective curves over a perfect fiel
d of characteristic $p$ such that $C_n$ is a branched cover of $C_{n-1}$ a
nd $C_n$ is a branched Galois $Z/(p^n)$-cover of $C_0$. For nice examples
of $Z_p$ towers\, the growth of the genus is stable: for sufficiently lar
ge $n$\, the genus of $C_n$ is a quadratic polynomial in $p^n$. In charac
teristic $p$\, there are additional curve invariants like the a-number whi
ch 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 s
table 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 registe
r here: \n\nhttps://ucsd.zoom.us/meeting/register/tJIuf-6uqjoiH9eVbtLN9y1
G9l0qEBmbDRmV\n
LOCATION:https://researchseminars.org/talk/POINT/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jef Laga (University of Cambridge)
DTSTART;VALUE=DATE-TIME:20210215T130000Z
DTEND;VALUE=DATE-TIME:20210215T133000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/27
DESCRIPTION:Title: S
elmer groups of some families of genus 3 curves and abelian surfaces\n
by Jef Laga (University of Cambridge) as part of POINT: New Developments i
n Number Theory\n\n\nAbstract\nManjul Bhargava and Arul Shankar have deter
mined the average size of the $n$-Selmer group of the family of all ellipt
ic curves over $\\mathbb{Q}$ ordered by height\, for $n$ at most $5$. In t
his talk we will consider a family of nonhyperelliptic genus $3$ curves\,
and bound the average size of the $2$-Selmer group of their Jacobians. Thi
s implies that a majority of curves in this family have relatively few rat
ional points. We also consider a family of abelian surfaces which are not
principally polarized and obtain similar results. The proof is a combinati
on of the theory of simple singularities\, graded Lie algebras and orbit-c
ounting techniques.\n
LOCATION:https://researchseminars.org/talk/POINT/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Clifton (Emory University)
DTSTART;VALUE=DATE-TIME:20210215T133000Z
DTEND;VALUE=DATE-TIME:20210215T140000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/28
DESCRIPTION:Title: A
n exponential bound for exponential diffsequences\nby Alexander Clifto
n (Emory University) as part of POINT: New Developments in Number Theory\n
\n\nAbstract\nA theorem of van der Waerden states that for any positive in
teger $r$\, if you partition $\\mathbf{N}$ into $r$ disjoint subsets\, the
n one of them will contain arbitrarily long arithmetic progressions. It is
natural to ask what other arithmetic structures are preserved when partit
ioning $\\mathbf{N}$ into a finite number of disjoint sets and to pose qua
ntitative questions about these. We consider $D$-diffsequences\, introduce
d by Landman and Robertson\, which are increasing sequences in which the c
onsecutive differences all lie in some given set $D$. Here\, we consider t
he case where $D$ consists of all powers of $2$ and define $f(k)$ to be th
e smallest $n$ such that partitioning $\\{1\,2\,\\cdots\,n\\}$ into $2$ su
bsets guarantees the presence of a $D$-diffsequence of length $k$ containe
d entirely within one subset. We establish that $f(k)$ grows exponentially
.\n
LOCATION:https://researchseminars.org/talk/POINT/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maryam Khaqan (Emory University)
DTSTART;VALUE=DATE-TIME:20210303T003000Z
DTEND;VALUE=DATE-TIME:20210303T010000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/29
DESCRIPTION:Title: E
lliptic Curves and Thompson's Sporadic Group\nby Maryam Khaqan (Emory
University) as part of POINT: New Developments in Number Theory\n\n\nAbstr
act\nMoonshine began as a series of numerical coincidences connecting fini
te groups to modular forms. It has since evolved into a rich theory that s
heds light on the underlying structures that these coincidences reflect.\n
\n\nWe prove the existence of one such structure\, a module for the Thomps
on group\, whose graded traces are specific half-integral weight weakly ho
lomorphic modular forms. We then proceed to use this module to study the r
anks 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://researchseminars.org/talk/POINT/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Teppei Takamatsu (University of Tokyo)
DTSTART;VALUE=DATE-TIME:20210303T010000Z
DTEND;VALUE=DATE-TIME:20210303T013000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/30
DESCRIPTION:Title: O
n finiteness of twisted forms of hyperkähler varieties\nby Teppei Tak
amatsu (University of Tokyo) as part of POINT: New Developments in Number
Theory\n\n\nAbstract\nFor a finite field extension $L/K$ and a variety $X$
over $K$\,\nlet $Tw_{L/K} (X)$ be the set of isomorphism classes of varie
ties $Y$\nover $K$ which are isomorphic to $X$ after the base change to $L
$ (i.e.\nthe set of twisted forms of $X$ via $L/K$). In this talk\, we pro
ve the\nfiniteness of $Tw_{L/K}$ for K3 surfaces of characteristic away fr
om 2\nand hyperkähler varieties of characteristic 0. This work is a\ngene
ralization of Cattaneo-Fu's work on real forms of hyperkähler\nvarieties.
We also give an application to the finiteness of derived\nequivalent twis
ted forms of hyperkähler varieties.\n
LOCATION:https://researchseminars.org/talk/POINT/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bella Tobin (Oklahoma State University)
DTSTART;VALUE=DATE-TIME:20210601T200000Z
DTEND;VALUE=DATE-TIME:20210601T203000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/33
DESCRIPTION:Title: R
eduction of post-critically finite polynomials\nby Bella Tobin (Oklaho
ma State University) as part of POINT: New Developments in Number Theory\n
\n\nAbstract\nPost-critically finite maps are described as dynamical analo
gs of CM Abelian Varieties. A CM abelian varieties over a number field $K$
has everywhere good reduction in some finite extension $L/K$. This motiva
tes us to ask the question: do PCF maps have good reduction? We can use a
particular family of maps\, dynamical Belyi polynomials\, to provide nece
ssary and sufficient conditions for a PCF polynomial of degree $d$ to have
potential good reduction at a prime $p$. This is joint work with Jacqueli
ne Anderson and Michelle Manes.\n
LOCATION:https://researchseminars.org/talk/POINT/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shaoyun Yi (University of South Carolina)
DTSTART;VALUE=DATE-TIME:20210601T203000Z
DTEND;VALUE=DATE-TIME:20210601T210000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/34
DESCRIPTION:Title: O
n counting cuspidal automorphic representations of GSp(4)\nby Shaoyun
Yi (University of South Carolina) as part of POINT: New Developments in Nu
mber Theory\n\n\nAbstract\nThere are some well-known classical equidistrib
ution results like Sato-Tate conjecture for elliptic curves and equidistri
bution of Hecke eigenvalues of elliptic cusp forms. In this talk\, we will
discuss a similar equidistribution result for a family of cuspidal automo
rphic representations for GSp(4). We formulate our theorem explicitly in t
erms of the number of cuspidal automorphic representations for GSp(4) with
certain conditions at the local places. To count the number of these cusp
idal automorphic representations\, we will explore the connection between
Siegel cusp forms of degree 2 and cuspidal automorphic representations of
GSp(4). This is a joint work with Manami Roy and Ralf Schmidt.\n
LOCATION:https://researchseminars.org/talk/POINT/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Battistoni (Universite de Framche-Comte)
DTSTART;VALUE=DATE-TIME:20210615T123000Z
DTEND;VALUE=DATE-TIME:20210615T130000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/35
DESCRIPTION:Title: O
n elliptic curves over $\\mathbb{Q}(T)$ and their ranks\nby Francesco
Battistoni (Universite de Framche-Comte) as part of POINT: New Development
s in Number Theory\n\n\nAbstract\nWe consider elliptic curves over $\\math
bb{Q}(T)$ admitting Weierstrass model with coefficients being polynomials
of small degree\, so that they are rational elliptic surfaces. In joint wo
rk with Sandro Bettin and Christophe Delaunay\, we apply Nagao's formula i
n order to detect the value of their ranks: this approach is orthogonal to
other geometric investigations\, and gives the values of the ranks by loo
king at purely algebraic properties like the factorization of some integer
polynomials. We also prove that\, whenever restricting to some specific f
amilies of curves\, the generic curve in these families has rank $0$.\n
LOCATION:https://researchseminars.org/talk/POINT/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geoffrey Akers (CUNY Graduate Center)
DTSTART;VALUE=DATE-TIME:20210615T130000Z
DTEND;VALUE=DATE-TIME:20210615T133000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/36
DESCRIPTION:Title: O
n a universal deformation ring that is a discrete valuation ring\nby G
eoffrey Akers (CUNY Graduate Center) as part of POINT: New Developments in
Number Theory\n\n\nAbstract\nWe consider a crystalline universal deformat
ion ring $R$ of an $n$-dimensional\, mod $p$ Galois representation whose s
emisimplification is the direct sum of two non-isomorphic absolutely irred
ucible representations. Under some hypotheses\, we obtain that $R$ is a di
screte valuation ring. The method examines the ideal of reducibility of $R
$\, which is used to construct extensions of representations in a Selmer g
roup with specified dimension. This can be used to deduce modularity of r
epresentations.\n
LOCATION:https://researchseminars.org/talk/POINT/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ross Paterson (University of Glasgow)
DTSTART;VALUE=DATE-TIME:20210628T163000Z
DTEND;VALUE=DATE-TIME:20210628T170000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/37
DESCRIPTION:Title: A
verage Ranks of Elliptic Curves after $p$-Extension\nby Ross Paterson
(University of Glasgow) as part of POINT: New Developments in Number Theor
y\n\n\nAbstract\nAs $E$ varies among elliptic curves defined over the rati
onal numbers\, a theorem of Bhargava and Shankar shows that the average ra
nk of the Mordell--Weil group $E(\\mathbb{Q})$ is bounded. If we fix a nu
mber 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 questio
ns for a restricted set of $K$.\n
LOCATION:https://researchseminars.org/talk/POINT/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damián Gvirtz (University College London)
DTSTART;VALUE=DATE-TIME:20210628T170000Z
DTEND;VALUE=DATE-TIME:20210628T173000Z
DTSTAMP;VALUE=DATE-TIME:20220128T031855Z
UID:POINT/38
DESCRIPTION:Title: T
he 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" rational
points? A possible formalisation of this notion is the (weak) Hilbert Prop
erty for algebraic varieties\, a generalisation of Hilbert's classical irr
educibility theorem. I will report on joint work in progress with G. Mezze
dimi about a conjecture due to Campana and Corvaja-Zannier which concerns
this property in the case of Enriques surfaces.\n
LOCATION:https://researchseminars.org/talk/POINT/38/
END:VEVENT
END:VCALENDAR