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BEGIN:VEVENT
SUMMARY:David Roberts (University of Minnesota\, Morris)
DTSTART:20220411T160000Z
DTEND:20220411T170000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/1
DESCRIPTION:Title: Modularity problems for hypergeometric motives\nby David Ro
berts (University of Minnesota\, Morris) as part of Explicit Methods for M
odularity\n\n\nAbstract\nI will set up a motivic inverse problem which ask
s whether there exists a nondegenerate motive\n$M$ having a given Hodge ve
ctor $h = (h^{w\,0}\,h^{w-1\,1}\,...\,h^{1\,w-1}\,h^{0\,w})$. \nThere is n
ot even an accepted conjectural description of the yes/no borderline. \nSo
it is a problem ripe for computational exploration.\n\nI'll briefly recal
l the theory of hypergeometric motives. A recent survey joint with\nFerna
ndo Rodriguez Villegas explains how this theory answers "yes" for many $h$
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 briefly in
troduced in the survey\, "semi hypergeometric motives". In this variant\,
many of the Hodge numbers can be zero. One thereby gets a "yes" respons
e 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 hyperg
eometric motives have relatively small conductors $N$\, facilitating expli
cit\nconnections with automorphic forms. I'll exhibit several connections
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 correspo
nding 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 organized
so that previous familiarity with motives is not an essential prerequisit
e.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kimball Martin (University of Oklahoma)
DTSTART:20220411T170000Z
DTEND:20220411T180000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/2
DESCRIPTION:Title: Counting abelian surfaces with RM\nby Kimball Martin (Unive
rsity of Oklahoma) as part of Explicit Methods for Modularity\n\n\nAbstrac
t\nClassical modularity gives a correspondence between rational elliptic c
urves\nand rational modular forms of weight 2. In particular\, it is inst
rumental\nin enumerating elliptic curves up to a given conductor. More ge
nerally\,\nmodularity relates rational abelian varieties with sufficient s
ymmetry\n(of GL(2) type) to weight 2 modular forms. I will talk about ong
oing joint\nwork with Alex Cowan towards counting rational abelian surface
s with \nreal multplication (RM). One perspective is to use the lens of m
odularity\, \nand another is to study rational points on Hilbert modular s
urfaces.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juanita Duque-Rosero (Dartmouth College)
DTSTART:20220412T200000Z
DTEND:20220412T203000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/4
DESCRIPTION:Title: Enumerating triangular modular curves of small genus\nby Ju
anita Duque-Rosero (Dartmouth College) as part of Explicit Methods for Mod
ularity\n\n\nAbstract\nTriangular modular curves are a generalization of m
odular curves that arise from quotients of the upper half-plane by congrue
nce subgroups of hyperbolic triangle groups. These curves arise from Belyi
maps with monodromy $\\operatorname{PGL}_2(\\mathbb{F}_q)$ or $\\operator
name{PSL}_2(\\mathbb{F}_q)$. In this talk\, we will present a computationa
l approach to enumerate all triangular modular curves of genus 0\, 1\, and
2. This is joint work with John Voight.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manami Roy (Fordham University)
DTSTART:20220412T203000Z
DTEND:20220412T210000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/5
DESCRIPTION:Title: Dimensions for the spaces of Siegel cusp forms of Klingen level
4\nby Manami Roy (Fordham University) as part of Explicit Methods for
Modularity\n\n\nAbstract\nMany mathematicians have studied dimension and
codimension formulas for the spaces of Siegel cusp forms of degree $2$. Th
e dimensions of the spaces of Siegel cusp forms of non-squarefree levels a
re mostly not available in the literature. This talk will present new dime
nsion formulas of Siegel cusp forms of degree $2$\, weight $k$\, and level
$4$ for two congruence subgroups. Our method relies on counting a particu
lar set of cuspidal automorphic representations of $\\operatorname{GSp}(4)
$ and exploring its connection to dimensions of spaces of Siegel cusp form
s of degree $2$. This work is joint with Ralf Schmidt and Shaoyun Yi.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (Brown University)
DTSTART:20220413T200000Z
DTEND:20220413T203000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/6
DESCRIPTION:Title: An introduction to open image theorems for abelian varieties\nby Isabel Vogt (Brown University) as part of Explicit Methods for Modul
arity\n\n\nAbstract\nIn this expository talk\, I will give a brief introdu
ction to open image theorems for abelian varieties over number fields\, fo
cusing on the case of elliptic curves over $\\mathbb{Q}$.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (Brown University)
DTSTART:20220413T203000Z
DTEND:20220413T210000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/7
DESCRIPTION:Title: Explicit determination of nonsurjective primes for abelian surf
aces\nby Isabel Vogt (Brown University) as part of Explicit Methods fo
r Modularity\n\n\nAbstract\nSerre proved that when the Jacobian $J$ of a g
enus 2 curve over $\\mathbb{Q}$ has typical endomorphism ring\, there is a
finite set of primes $\\ell$ for which the Galois action on the $\\ell$-t
orsion of $J$ is not all of $\\text{GSp}_4(\\mathbb{F}_\\ell)$. In this t
alk I will report on joint work with Barinder Banwait\, Armand Brumer\, Hy
un Jong Kim\, Zev Klagsbrun\, Jacob Mayle\, and Padmavathi Srinivasan on t
he problem of explicitly finding this finite set. In the course of our wo
rk\, based on an algorithm of Dieulefait\, we explicitly use Serre's Conje
cture (now a theorem of Khare--Wintenberger) on the modularity of odd two-
dimensional Galois representations.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Love (McGill University)
DTSTART:20220413T210000Z
DTEND:20220413T213000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/8
DESCRIPTION:Title: Computing cusp forms over function fields\nby Jonathan Love
(McGill University) as part of Explicit Methods for Modularity\n\n\nAbstr
act\nThere is a vast collection of literature and computational tools avai
lable for modular forms over number fields\, but the function field case i
s comparatively less well understood\, and far fewer examples have been ge
nerated. In this talk\, I will summarize an algorithm that can be used to
compute a space of everywhere unramified cusp forms over the function fiel
d of a curve $X$ over $\\mathbb{F}_p$\, and discuss some outputs of the al
gorithm\, implications\, and related questions.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Viray (University of Washington)
DTSTART:20220414T173000Z
DTEND:20220414T180000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/9
DESCRIPTION:Title: Quadratic points on intersections of quadrics\nby Bianca Vi
ray (University of Washington) as part of Explicit Methods for Modularity\
n\n\nAbstract\nA projective degree $d$ variety always has a point defined
over a degree $d$ field extension. For many degree $d$ varieties\, this i
s the best possible statement\, that is\, there exist classes of degree $d
$ varieties that never have points over extensions of degree less than $d$
(nor even over extensions whose degree is nonzero modulo $d$). However\,
there are some classes of degree $d$ varieties that obtain points over ex
tensions of smaller degree\, for example\, degree $9$ surfaces in $\\mathb
b{P}^9$\, and $6$-dimensional intersections of quadrics over local fields.
In this talk\, we explore this question for intersections of quadrics.
In particular\, we prove that a smooth complete intersection of two quadri
cs 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 join
t work with Brendan Creutz.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Holly Swisher (Oregon State University)
DTSTART:20220414T180000Z
DTEND:20220414T190000Z
DTSTAMP:20260422T212527Z
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 fascinatin
g infinite series representations of $1/\\pi$. In the 1980's it was dete
rmined that these series provided efficient means for approximating $\\pi$
. Since then discovering and proving series of this type have been of int
erest\, and a variety of techniques have been used. Motivated by work of
Chan\, Chan\, and Liu\, we obtain a new general theorem yielding corollari
es that produce generalized Ramanujan-Sato series for $1/\\pi$. We use the
se corollaries to construct explicit examples arising from modular forms o
n arithmetic triangle groups. This work is joint with Angelica Babei\, Le
a Beneish\, Manami Roy\, Bella Tobin\, and Fang-Ting Tu. It was initiated
as part of the Women in Numbers 5 workshop.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Breen (Clemson University)
DTSTART:20220414T190000Z
DTEND:20220414T193000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/11
DESCRIPTION:Title: Computing Hilbert modular forms via a trace formula\nby Be
njamin Breen (Clemson University) as part of Explicit Methods for Modulari
ty\n\n\nAbstract\nWe present an explicit method for computing spaces of Hi
lbert modular forms using a trace formula. We describe the main algorithmi
c challenges and discuss the advantages and shortcomings of this method in
comparison to other methods for producing Hilbert modular forms. We concl
ude with computations.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Kieffer (Harvard University)
DTSTART:20220415T170000Z
DTEND:20220415T173000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/12
DESCRIPTION:Title: Asymptotically faster point counting on abelian surfaces\n
by Jean Kieffer (Harvard University) as part of Explicit Methods for Modul
arity\n\n\nAbstract\nThe point counting problem asks\, given an abelian va
riety $A$ of dimension $g$ over a finite field $\\mathbb{F}_q$\, to comput
e the characteristic polynomial of Frobenius on $A$. In large characterist
ic\, the classical approach to this problem is to apply Schoof's algorithm
and study the action of Frobenius on $\\ell$-torsion subgroups\; in the c
ase of elliptic curves\, a further improvement by Elkies consists in repla
cing 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 dimensions\,
and specifically to obtain asymptotically faster point counting algorithm
s for principally polarized abelian surfaces. As a key step we show that i
sogenies between p.p. abelian surfaces can be efficiently computed from hi
gher-dimensional modular equations\, both in theory and in practice.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiva Chidambaram (MIT)
DTSTART:20220415T173000Z
DTEND:20220415T183000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/13
DESCRIPTION:Title: Modularity of typical abelian surfaces over $\\mathbb{Q}$\
nby Shiva Chidambaram (MIT) as part of Explicit Methods for Modularity\n\n
\nAbstract\nThe modularity lifting theorem of Boxer-Calegari-Gee-Pilloni e
stablished for the first time the existence of infinitely many modular abe
lian surfaces $A / \\Q$ upto twist with $\\End_{\\C}(A) = \\Z$. We render
this explicit by first finding some abelian surfaces whose associated mod-
$p$ representation is residually modular and for which the modularity lift
ing theorem is applicable\, and then transferring modularity in a family o
f abelian surfaces with fixed $3$-torsion representation. Let $\\rho: G_{\
\Q} \\rightarrow \\GSp(4\,\\F_3)$ be a Galois representation with cyclotom
ic similitude character. Then\, the transfer of modularity happens in the
moduli space of genus $2$ curves $C$ such that $C$ has a rational Weierstr
ass 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://researchseminars.org/talk/ExplicitModularity/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Voight (Dartmouth College)
DTSTART:20220415T183000Z
DTEND:20220415T190000Z
DTSTAMP:20260422T212527Z
UID:ExplicitModularity/14
DESCRIPTION:Title: Sato-Tate groups and modularity for atypical abelian surfaces<
/a>\nby John Voight (Dartmouth College) as part of Explicit Methods for Mo
dularity\n\n\nAbstract\nWe discuss in detail what it means for an abelian
surface $A$ over a number field to be modular\, organizing conjectures and
theorems that associate to $A$ a modular form with matching $L$-function.
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 classi
cal\, Bianchi\, or Hilbert modular forms\; and for each possibility\, we p
rovide a genus 2 curve with small conductor from which it arises. This is
joint work with Andrew Booker\, Jeroen Sijsling\, Andrew Sutherland\, and
Dan Yasaki.\n
LOCATION:https://researchseminars.org/talk/ExplicitModularity/14/
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