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BEGIN:VEVENT
SUMMARY:David Roberts (University of Minnesota\, Morris)
DTSTART:20220411T160000Z
DTEND:20220411T170000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/1
DESCRIPTION:Title: Modularity problems for hypergeometric motives\nby D
avid Roberts (University of Minnesota\, Morris) as part of Explicit Method
s for Modularity\n\n\nAbstract\nI will set up a motivic inverse problem wh
ich asks whether there exists a nondegenerate motive\n$M$ having a given H
odge vector $h = (h^{w\,0}\,h^{w-1\,1}\,...\,h^{1\,w-1}\,h^{0\,w})$. \nThe
re is not even an accepted conjectural description of the yes/no borderlin
e. \nSo it is a problem ripe for computational exploration.\n\nI'll briefl
y recall the theory of hypergeometric motives. A recent survey joint with
\nFernando Rodriguez Villegas explains how this theory answers "yes" for m
any $h$ with\nall entries positive\, including all such $h$ with $\\sum h^
{p\,q} \\leq 21$.\n\nThe main part of the talk will be about a variant bri
efly introduced in the survey\, "semi hypergeometric motives". In this va
riant\, many of the Hodge numbers can be zero. One thereby gets a "yes"
response for all sorts of Hodge vectors\, as I'll illustrate with $h=(2\,0
\,1\,0\,0\,0\,0\,1\,0\,2)$ and $h=(2\,1\,0\,1\,0\,0\,1\,0\,1\,2)$.\n\nSemi
hypergeometric motives have relatively small conductors $N$\, facilitatin
g explicit\nconnections with automorphic forms. I'll exhibit several conn
ections in a classical\ncontext\, including one with $h=(1\,0\,0\,0\,0\,0\
,0\,0\,0\,1)$ and $N=4$. I'll exhibit\nseveral examples where finding a c
orresponding automorphic form seems plausible\,\nincluding one with $h = (
1\,0\,1\,0\,1\,0\,0\,1\,0\,1\,0\,1)$ and $N=2^7 3$.\n\nThe talk will be or
ganized so that previous familiarity with motives is not an essential prer
equisite.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kimball Martin (University of Oklahoma)
DTSTART:20220411T170000Z
DTEND:20220411T180000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/2
DESCRIPTION:Title: Counting abelian surfaces with RM\nby Kimball Martin
(University of Oklahoma) as part of Explicit Methods for Modularity\n\n\n
Abstract\nClassical modularity gives a correspondence between rational ell
iptic curves\nand rational modular forms of weight 2. In particular\, it
is instrumental\nin enumerating elliptic curves up to a given conductor.
More generally\,\nmodularity relates rational abelian varieties with suffi
cient symmetry\n(of GL(2) type) to weight 2 modular forms. I will talk ab
out ongoing joint\nwork with Alex Cowan towards counting rational abelian
surfaces with \nreal multplication (RM). One perspective is to use the le
ns of modularity\, \nand another is to study rational points on Hilbert mo
dular surfaces.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juanita Duque-Rosero (Dartmouth College)
DTSTART:20220412T200000Z
DTEND:20220412T203000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/4
DESCRIPTION:Title: Enumerating triangular modular curves of small genus
\nby Juanita Duque-Rosero (Dartmouth College) as part of Explicit Methods
for Modularity\n\n\nAbstract\nTriangular modular curves are a generalizati
on of modular curves that arise from quotients of the upper half-plane by
congruence subgroups of hyperbolic triangle groups. These curves arise fro
m Belyi maps with monodromy $\\operatorname{PGL}_2(\\mathbb{F}_q)$ or $\\o
peratorname{PSL}_2(\\mathbb{F}_q)$. In this talk\, we will present a compu
tational approach to enumerate all triangular modular curves of genus 0\,
1\, and 2. This is joint work with John Voight.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manami Roy (Fordham University)
DTSTART:20220412T203000Z
DTEND:20220412T210000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/5
DESCRIPTION:Title: Dimensions for the spaces of Siegel cusp forms of Klinge
n level 4\nby Manami Roy (Fordham University) as part of Explicit Meth
ods for Modularity\n\n\nAbstract\nMany mathematicians have studied dimensi
on and codimension formulas for the spaces of Siegel cusp forms of degree
$2$. The dimensions of the spaces of Siegel cusp forms of non-squarefree l
evels are mostly not available in the literature. This talk will present n
ew dimension formulas of Siegel cusp forms of degree $2$\, weight $k$\, an
d level $4$ for two congruence subgroups. Our method relies on counting a
particular set of cuspidal automorphic representations of $\\operatorname{
GSp}(4)$ and exploring its connection to dimensions of spaces of Siegel cu
sp forms of degree $2$. This work is joint with Ralf Schmidt and Shaoyun Y
i.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (Brown University)
DTSTART:20220413T200000Z
DTEND:20220413T203000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/6
DESCRIPTION:Title: An introduction to open image theorems for abelian varie
ties\nby Isabel Vogt (Brown University) as part of Explicit Methods fo
r Modularity\n\n\nAbstract\nIn this expository talk\, I will give a brief
introduction to open image theorems for abelian varieties over number fiel
ds\, focusing on the case of elliptic curves over $\\mathbb{Q}$.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (Brown University)
DTSTART:20220413T203000Z
DTEND:20220413T210000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/7
DESCRIPTION:Title: Explicit determination of nonsurjective primes for abeli
an surfaces\nby Isabel Vogt (Brown University) as part of Explicit Met
hods for Modularity\n\n\nAbstract\nSerre proved that when the Jacobian $J$
of a genus 2 curve over $\\mathbb{Q}$ has typical endomorphism ring\, the
re is a finite set of primes $\\ell$ for which the Galois action on the $\
\ell$-torsion of $J$ is not all of $\\text{GSp}_4(\\mathbb{F}_\\ell)$. In
this talk I will report on joint work with Barinder Banwait\, Armand Brum
er\, Hyun Jong Kim\, Zev Klagsbrun\, Jacob Mayle\, and Padmavathi Srinivas
an on the problem of explicitly finding this finite set. In the course of
our work\, based on an algorithm of Dieulefait\, we explicitly use Serre'
s Conjecture (now a theorem of Khare--Wintenberger) on the modularity of o
dd two-dimensional Galois representations.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Love (McGill University)
DTSTART:20220413T210000Z
DTEND:20220413T213000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/8
DESCRIPTION:Title: Computing cusp forms over function fields\nby Jonath
an Love (McGill University) as part of Explicit Methods for Modularity\n\n
\nAbstract\nThere is a vast collection of literature and computational too
ls available for modular forms over number fields\, but the function field
case is comparatively less well understood\, and far fewer examples have
been generated. In this talk\, I will summarize an algorithm that can be u
sed to compute a space of everywhere unramified cusp forms over the functi
on field of a curve $X$ over $\\mathbb{F}_p$\, and discuss some outputs of
the algorithm\, implications\, and related questions.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART:20220414T173000Z
DTEND:20220414T180000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/9
DESCRIPTION:Title: Quadratic points on intersections of quadrics\nby Bi
anca Viray (University of Washington) as part of Explicit Methods for Modu
larity\n\n\nAbstract\nA projective degree $d$ variety always has a point d
efined over a degree $d$ field extension. For many degree $d$ varieties\,
this is the best possible statement\, that is\, there exist classes of de
gree $d$ varieties that never have points over extensions of degree less t
han $d$ (nor even over extensions whose degree is nonzero modulo $d$). Ho
wever\, there are some classes of degree $d$ varieties that obtain points
over extensions of smaller degree\, for example\, degree $9$ surfaces in $
\\mathbb{P}^9$\, and $6$-dimensional intersections of quadrics over local
fields. In this talk\, we explore this question for intersections of quad
rics. In particular\, we prove that a smooth complete intersection of two
quadrics of dimension at least $2$ over a number field has index dividing
$2$\, i.e.\, that it possesses a rational $0$-cycle of degree $2$. This
is joint work with Brendan Creutz.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Swisher (Oregon State University)
DTSTART:20220414T180000Z
DTEND:20220414T190000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/10
DESCRIPTION:Title: Generalized Ramanujan-Sato series arising from modular
forms\nby Holly Swisher (Oregon State University) as part of Explicit
Methods for Modularity\n\n\nAbstract\nIn 1914\, Ramanujan gave several fas
cinating infinite series representations of $1/\\pi$. In the 1980's it w
as determined that these series provided efficient means for approximating
$\\pi$. Since then discovering and proving series of this type have been
of interest\, and a variety of techniques have been used. Motivated by w
ork of Chan\, Chan\, and Liu\, we obtain a new general theorem yielding co
rollaries that produce generalized Ramanujan-Sato series for $1/\\pi$. We
use these corollaries to construct explicit examples arising from modular
forms on arithmetic triangle groups. This work is joint with Angelica Bab
ei\, Lea Beneish\, Manami Roy\, Bella Tobin\, and Fang-Ting Tu. It was ini
tiated as part of the Women in Numbers 5 workshop.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Breen (Clemson University)
DTSTART:20220414T190000Z
DTEND:20220414T193000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/11
DESCRIPTION:Title: Computing Hilbert modular forms via a trace formula
\nby Benjamin Breen (Clemson University) as part of Explicit Methods for M
odularity\n\n\nAbstract\nWe present an explicit method for computing space
s of Hilbert modular forms using a trace formula. We describe the main alg
orithmic challenges and discuss the advantages and shortcomings of this me
thod in comparison to other methods for producing Hilbert modular forms. W
e conclude with computations.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Kieffer (Harvard University)
DTSTART:20220415T170000Z
DTEND:20220415T173000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/12
DESCRIPTION:Title: Asymptotically faster point counting on abelian surface
s\nby Jean Kieffer (Harvard University) as part of Explicit Methods fo
r Modularity\n\n\nAbstract\nThe point counting problem asks\, given an abe
lian variety $A$ of dimension $g$ over a finite field $\\mathbb{F}_q$\, to
compute the characteristic polynomial of Frobenius on $A$. In large chara
cteristic\, the classical approach to this problem is to apply Schoof's al
gorithm and study the action of Frobenius on $\\ell$-torsion subgroups\; i
n the case of elliptic curves\, a further improvement by Elkies consists i
n replacing the full $\\ell$-torsion by the (cyclic) kernel of an $\\ell$-
isogeny. The aim of this talk is to extend Elkies's method to higher dimen
sions\, and specifically to obtain asymptotically faster point counting al
gorithms for principally polarized abelian surfaces. As a key step we show
that isogenies between p.p. abelian surfaces can be efficiently computed
from higher-dimensional modular equations\, both in theory and in practice
.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART:20220415T173000Z
DTEND:20220415T183000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/13
DESCRIPTION:Title: Modularity of typical abelian surfaces over $\\mathbb{Q
}$\nby Shiva Chidambaram (MIT) as part of Explicit Methods for Modular
ity\n\n\nAbstract\nThe modularity lifting theorem of Boxer-Calegari-Gee-Pi
lloni established for the first time the existence of infinitely many modu
lar abelian surfaces $A / \\Q$ upto twist with $\\End_{\\C}(A) = \\Z$. We
render this explicit by first finding some abelian surfaces whose associat
ed mod-$p$ representation is residually modular and for which the modulari
ty lifting theorem is applicable\, and then transferring modularity in a f
amily of abelian surfaces with fixed $3$-torsion representation. Let $\\rh
o: G_{\\Q} \\rightarrow \\GSp(4\,\\F_3)$ be a Galois representation with c
yclotomic similitude character. Then\, the transfer of modularity happens
in the moduli space of genus $2$ curves $C$ such that $C$ has a rational W
eierstrass point and $\\mathrm{Jac}(C)[3] \\simeq \\rho$. Using invariant
theory\, we find explicit parametrization of the universal curve over this
space.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART:20220415T183000Z
DTEND:20220415T190000Z
DTSTAMP:20260404T093754Z
UID:ExplicitModularity/14
DESCRIPTION:Title: Sato-Tate groups and modularity for atypical abelian su
rfaces\nby John Voight (Dartmouth College) as part of Explicit Methods
for Modularity\n\n\nAbstract\nWe discuss in detail what it means for an a
belian surface $A$ over a number field to be modular\, organizing conjectu
res and theorems that associate to $A$ a modular form with matching $L$-fu
nction. The explicit description of this modular form depends on the real
Galois endomorphism type of $A$\, or equivalently on its Sato–Tate group
. For $A$ defined over the rational numbers\, this description can involve
classical\, Bianchi\, or Hilbert modular forms\; and for each possibility
\, we provide a genus 2 curve with small conductor from which it arises. T
his is joint work with Andrew Booker\, Jeroen Sijsling\, Andrew Sutherland
\, and Dan Yasaki.\n
LOCATION:https://stable.researchseminars.org/talk/ExplicitModularity/14/
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