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BEGIN:VEVENT
SUMMARY:Richard Hain (Duke University)
DTSTART:20210510T140000Z
DTEND:20210510T150000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/1
DESCRIPTION:Title: Weighted completion of Galois groups and rational points\nby Richar
d Hain (Duke University) as part of Rational Points and Galois Representat
ions\n\n\nAbstract\nThe goal of this talk is to explain how one can use we
ighted\nand relative completion to generalize some recent results (arXiv:2
010.07331) of Li\, Litt\,\nSalter and Srinivasan. In particular\, we will
explain how algebraic\ncompletions of mapping class groups and their arith
metic analogues can\nbe used to give examples\, for each $g > 3$\, $n \\ge
0$\, $r > 0$\, of a smooth\nprojective curve of genus $g > 3$ over a fini
tely generated field $K$ of\nchar 0 where $\\#C(K) = n$\, $\\mathrm{Pic}^1
(C)(K)$ is non-empty and $\\mathrm{Pic}^0(C)(K)$\ncontains a free abelian
subgroup of rank $n + r - 1$.\n\nThe talk will begin with a review of how
one uses weighted (unipotent)\ncompletion of the Galois group of the funct
ion field of a moduli\nspaces of curves (plus other structure) to study ra
tional points of\nthe the universal curve over its generic point. If there
is sufficient\ntime\, I will explain how this leads to a theory of charac
teristic\nclasses of rational points.\n\nREFERENCES:\n\n[HM2003] R. Hain\,
M. Matsumoto: Weighted completion of Galois groups\n and Galois a
ctions on the fundamental group of P^1-{0\,1\,infty}.\n Compositio
Math. 139 (2003)\, 119--167.\n arXiv:math/0006158\n\n[H2011] R.
Hain: Rational points of universal curves\, J. Amer. Math. Soc. 24 (2011)\
,\n 709--769.\n https://www.ams.org/journals/jams/2011-24-
03/S0894-0347-2011-00693-0/home.html\n\n[LLSP] W. Li\, D. Litt\, N. Salt
er\, P. Srinivasan: Surface bundles and the section\n conjecture\,
arXiv:2010.07331.\n\n[W2019] T. Watanabe: Rational points of universal c
urves in positive characteristics\,\n Trans. Amer. Math. Soc. 372
(2019)\, 7639--7676.\n https://www.ams.org/journals/tran/2019-372-
11/S0002-9947-2019-07842-X/\n
LOCATION:https://researchseminars.org/talk/DioGal2021/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (University of Georgia)
DTSTART:20210510T153000Z
DTEND:20210510T163000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/2
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieties\
nby Padmavathi Srinivasan (University of Georgia) as part of Rational Poin
ts and Galois Representations\n\n\nAbstract\np-adic heights have been a ri
ch source of explicit functions vanishing on rational points on a curve. I
n this talk\, we will outline a new construction of canonical p-adic heigh
ts on abelian varieties from p-adic adelic metrics\, using p-adic Arakelov
theory developed by Besser. This construction closely mirrors Zhang's con
struction of canonical real valued heights from real-valued adelic metrics
. We will use this new construction to give direct explanations (avoiding
p-adic Hodge theory) of the key properties of height pairings needed for t
he quadratic Chabauty method for rational points. This is joint work in pr
ogress with Amnon Besser and Steffen Mueller.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kirsten Wickelgren (Duke University)
DTSTART:20210510T170000Z
DTEND:20210510T180000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/3
DESCRIPTION:Title: Colloquium Presentation: zeta functions and a quadratic enrichment\
nby Kirsten Wickelgren (Duke University) as part of Rational Points and Ga
lois Representations\n\n\nAbstract\nThe beautiful Weil conjectures connect
the Betti numbers of a complex variety whose defining equations can be re
duced mod p to the number of solutions mod p. We will discuss these connec
tions\, introduce A1-homotopy theory\, and an analogue in A1-homotopy theo
ry. The new work in this talk is joint with Margaret Bilu\, Wei Ho\, Padma
vathi Srinivasan\, and Isabel Vogt.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesca Bianchi (University of Groningen)
DTSTART:20210511T133000Z
DTEND:20210511T143000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/4
DESCRIPTION:Title: p-adic heights on Jacobians of genus 2 curves and applications\nby
Francesca Bianchi (University of Groningen) as part of Rational Points and
Galois Representations\n\n\nAbstract\nWe describe an algorithmic construc
tion of a p-adic height on the Jacobian of a genus 2 curve over the ration
als (here p is not necessarily of good reduction). In particular\, the foc
us will be on the local component at p of the height\, which is defined in
terms of some p-adic sigma/theta functions.\n\nThese local heights differ
from those in the Coleman--Gross construction in a crucial way\; neverthe
less\, in some cases we can prove a suitable comparison theorem. Thus\, we
can use our heights as an alternative to the Coleman--Gross heights in so
me instances of the quadratic Chabauty method. The application given in th
is talk concerns the rational points on some quite special genus 4 hyperel
liptic curves.\n\nThis talk is partly based on joint work with Enis Kaya a
nd Steffen Müller.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Betts (Harvard University)
DTSTART:20210511T145000Z
DTEND:20210511T155000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/5
DESCRIPTION:Title: Weights of Coleman functions and effective Chabauty--Kim\nby Alexan
der Betts (Harvard University) as part of Rational Points and Galois Repre
sentations\n\n\nAbstract\nThe Chabauty--Kim method is a technique for stud
ying the rational points on a curve X using motivic properties of quotient
s U of the fundamental group of X. For specific quotients U\, the method h
as been made effective in work of Coleman and later by Balakrishnan--Dogra
\, in the sense that it provides an explicit upper bound on the number of
rational points. In this talk\, I will discuss a recent project in which I
extend these effective results to all quotients U\, and give some applica
tions (joint work with David Corwin\, in progress) towards uniformity resu
lts for higher genus curves. A significant part of the proof\, which I wil
l discuss in more detail\, lies in defining a notion of "weight" for Colem
an analytic functions\, and showing\, following arguments of Balakrishnan-
-Dogra\, that the number of zeroes of a non-zero Coleman analytic function
can be bounded in terms of its weight.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Matschke (Boston University)
DTSTART:20210511T161000Z
DTEND:20210511T163000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/6
DESCRIPTION:Title: A general S-unit equation solver and tables of elliptic curves over num
ber fields\nby Benjamin Matschke (Boston University) as part of Ration
al Points and Galois Representations\n\n\nAbstract\nIn this talk we presen
t work in progress on a new highly optimized\nsolver for general and const
raint S-unit equations over number fields.\nIt has diophantine application
s including asymptotic Fermat theorems\,\nSiegel's method for computing in
tegral points\, and most strikingly for\ncomputing large tables of ellipti
c curves over number fields with good\nreduction outside given sets of pri
mes S. For the latter\, we improved\non the method of Koutsianas (Parshin\
, Shafarevich\, Elkies).\n
LOCATION:https://researchseminars.org/talk/DioGal2021/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jackson Morrow (Université de Montréal)
DTSTART:20210511T174000Z
DTEND:20210511T184000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/7
DESCRIPTION:Title: Progress on Mazur's Program B -- a horizontal perspective\nby Jacks
on Morrow (Université de Montréal) as part of Rational Points and Galois
Representations\n\n\nAbstract\nIn this talk\, I will discuss recent progr
ess on "Mazur's Program B" --- the problem of classifying all possibilitie
s for the image of Galois for an elliptic curve over $\\mathbb{Q}$. I will
focus on the horizontal perspective of Mazur's Program B\, which strives
to classify the composite (non-prime power) images of Galois for an ellipt
ic curve over $\\mathbb{Q}$. In particular\, I will introduce the notion o
f an entanglement of division fields\, give a group theoretic characteriza
tion of an entanglement\, and describe two sets of joint work. The first i
s with Harris Daniels where we classify all infinite families of elliptic
curves over $\\mathbb{Q}$ which have an "unexplained" entanglement between
their $p$ and $q$ division fields where $p\,q$ are distinct primes\, and
the second is with Harris Daniels and Álvaro Lozano-Robledo where we prov
e several results on elliptic curves (and more generally\, principally pol
arized abelian varieties) over $\\mathbb{Q}$ when the entanglement occurs
over an abelian extension.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbey Bourdon (Wake Forest University)
DTSTART:20210512T140000Z
DTEND:20210512T150000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/8
DESCRIPTION:Title: Families of Sporadic Points on Modular Curves\nby Abbey Bourdon (Wa
ke Forest University) as part of Rational Points and Galois Representation
s\n\n\nAbstract\nA closed point $x$ on a curve $C$ is sporadic if there ar
e only finitely many points of degree at most deg($x$). In the case where
$C$ is the modular curve $X_1(N)$\, a non-cuspidal sporadic point correspo
nds to an elliptic curve with a point of order $N$ defined over a number f
ield of unusually low degree. In this talk\, we will focus on sporadic poi
nts arising from $\\mathbb{Q}$-curves\, which are elliptic curves isogenou
s to their Galois conjugates. In particular\, our investigations are inspi
red by the following question: Are there only finitely many non-CM $\\math
bb{Q}$-curves which produce sporadic points on any modular curve of
the form $X_1(N)$? I will show that an affirmative answer to this questio
n would imply Serre's Uniformity Conjecture and discuss partial progress i
n the case of sporadic points of odd degree. This is joint work with Filip
Najman.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia University)
DTSTART:20210512T153000Z
DTEND:20210512T163000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/9
DESCRIPTION:Title: The Shafarevich conjecture for hypersurfaces in abelian varieties\n
by Will Sawin (Columbia University) as part of Rational Points and Galois
Representations\n\n\nAbstract\nFaltings proved the statement\, previously
conjectured by \nShafarevich\, that there are finitely many abelian variet
ies of \ndimension $n$\, defined over a fixed number field\, with good red
uction \noutside a fixed finite set of primes\, up to isomorphism. In join
t work \nwith Brian Lawrence\, we prove an analogous finiteness statement
for \nhypersurfaces in a fixed abelian variety with good reduction outside
a \nfinite set of primes. I will give an introduction to some of the idea
s \nin the proof\, which builds on $p$-adic Hodge theory techniques from w
ork \nof Lawrence and Venkatesh as well as a little-known area of algebrai
c \ngeometry\n
LOCATION:https://researchseminars.org/talk/DioGal2021/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (moderator) (Emory University)
DTSTART:20210512T170000Z
DTEND:20210512T180000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/10
DESCRIPTION:Title: Problem discussion session\nby David Zureick-Brown (moderator) (Em
ory University) as part of Rational Points and Galois Representations\n\n\
nAbstract\nThis problem discussion session features advance contrib
utions (pdf) from \n\n
LOCATION:https://researchseminars.org/talk/DioGal2021/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steffen Müller (University of Groningen)
DTSTART:20210511T163000Z
DTEND:20210511T165000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/11
DESCRIPTION:Title: Algorithms for quadratic Chabauty\nby Steffen Müller (University
of Groningen) as part of Rational Points and Galois Representations\n\n\nA
bstract\nQuadratic Chabauty is the simplest non-trivial instance of Kim's
non-abelian extension of Chabauty's method. Using p-adic heights\, it can
sometimes be made sufficiently explicit to compute the rational points on
certain modular curves. I will discuss a magma-implementation of this meth
od and show how to use it in practice. This is joint work with Jennifer Ba
lakrishnan\, Netan Dogra\, Jan Tuitman and Jan Vonk.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (University of Washington)
DTSTART:20210511T165000Z
DTEND:20210511T171000Z
DTSTAMP:20260422T212900Z
UID:DioGal2021/12
DESCRIPTION:Title: Computing non-surjective primes for abelian surfaces\nby Isabel Vo
gt (University of Washington) as part of Rational Points and Galois Repres
entations\n\n\nAbstract\nLet $A$ be a principally polarized abelian surfac
e over the rational numbers. Serre proved that there are finitely many pr
imes ell for which the Galois action on the ell-torsion points of $A$ is n
ot the entire group of symplectic similitudes $\\mathrm{GSp}_4(\\mathbb{F}
_\\ell)$. Later\, Dieulefait showed\, conditional on Serre's conjecture (
now a theorem of Khare and Wintenberger)\, that this finite set is effecti
vely computable. I will report on on-going joint work with Banwait\, Brum
er\, Kim\, Klagsbrun\, Mayle and Srinivasan where we implement this algori
thm and use it to compute nonsurjective primes for all genus 2 curves in t
he LMFDB.\n
LOCATION:https://researchseminars.org/talk/DioGal2021/12/
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