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SUMMARY:Richard Hain (Duke University)
DTSTART:20210510T140000Z
DTEND:20210510T150000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/1
DESCRIPTION:Title: Weighted completion of Galois groups and rational points\nby
Richard Hain (Duke University) as part of Rational Points and Galois Repr
esentations\n\n\nAbstract\nThe goal of this talk is to explain how one can
use weighted\nand relative completion to generalize some recent results (
arXiv:2010.07331) of Li\, Litt\,\nSalter and Srinivasan. In particular\, w
e will explain how algebraic\ncompletions of mapping class groups and thei
r arithmetic 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 finitely 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 a
belian 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 th
e function field of a moduli\nspaces of curves (plus other structure) to s
tudy rational points of\nthe the universal curve over its generic point. I
f there is sufficient\ntime\, I will explain how this leads to a theory of
characteristic\nclasses of rational points.\n\nREFERENCES:\n\n[HM2003] R.
Hain\, M. Matsumoto: Weighted completion of Galois groups\n and G
alois actions on the fundamental group of P^1-{0\,1\,infty}.\n Com
positio Math. 139 (2003)\, 119--167.\n arXiv:math/0006158\n\n[H201
1] R. Hain: Rational points of universal curves\, J. Amer. Math. Soc. 24
(2011)\,\n 709--769.\n https://www.ams.org/journals/jams/2
011-24-03/S0894-0347-2011-00693-0/home.html\n\n[LLSP] W. Li\, D. Litt\,
N. Salter\, P. Srinivasan: Surface bundles and the section\n conje
cture\, arXiv:2010.07331.\n\n[W2019] T. Watanabe: Rational points of univ
ersal curves in positive characteristics\,\n Trans. Amer. Math. So
c. 372 (2019)\, 7639--7676.\n https://www.ams.org/journals/tran/20
19-372-11/S0002-9947-2019-07842-X/\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Padmavathi Srinivasan (University of Georgia)
DTSTART:20210510T153000Z
DTEND:20210510T163000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/2
DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieti
es\nby Padmavathi Srinivasan (University of Georgia) as part of Ration
al Points and Galois Representations\n\n\nAbstract\np-adic heights have be
en a rich source of explicit functions vanishing on rational points on a c
urve. In this talk\, we will outline a new construction of canonical p-adi
c heights on abelian varieties from p-adic adelic metrics\, using p-adic A
rakelov theory developed by Besser. This construction closely mirrors Zhan
g's construction of canonical real valued heights from real-valued adelic
metrics. We will use this new construction to give direct explanations (av
oiding p-adic Hodge theory) of the key properties of height pairings neede
d for the quadratic Chabauty method for rational points. This is joint wor
k in progress with Amnon Besser and Steffen Mueller.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kirsten Wickelgren (Duke University)
DTSTART:20210510T170000Z
DTEND:20210510T180000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/3
DESCRIPTION:Title: Colloquium Presentation: zeta functions and a quadratic enrichme
nt\nby Kirsten Wickelgren (Duke University) as part of Rational Points
and Galois Representations\n\n\nAbstract\nThe beautiful Weil conjectures
connect the Betti numbers of a complex variety whose defining equations ca
n be reduced mod p to the number of solutions mod p. We will discuss these
connections\, introduce A1-homotopy theory\, and an analogue in A1-homoto
py theory. The new work in this talk is joint with Margaret Bilu\, Wei Ho\
, Padmavathi Srinivasan\, and Isabel Vogt.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesca Bianchi (University of Groningen)
DTSTART:20210511T133000Z
DTEND:20210511T143000Z
DTSTAMP:20260404T095118Z
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 Poi
nts and Galois Representations\n\n\nAbstract\nWe describe an algorithmic c
onstruction of a p-adic height on the Jacobian of a genus 2 curve over the
rationals (here p is not necessarily of good reduction). In particular\,
the focus will be on the local component at p of the height\, which is def
ined 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\; n
evertheless\, in some cases we can prove a suitable comparison theorem. Th
us\, we can use our heights as an alternative to the Coleman--Gross height
s in some instances of the quadratic Chabauty method. The application give
n in this talk concerns the rational points on some quite special genus 4
hyperelliptic curves.\n\nThis talk is partly based on joint work with Enis
Kaya and Steffen Müller.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Betts (Harvard University)
DTSTART:20210511T145000Z
DTEND:20210511T155000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/5
DESCRIPTION:Title: Weights of Coleman functions and effective Chabauty--Kim\nby
Alexander Betts (Harvard University) as part of Rational Points and Galoi
s Representations\n\n\nAbstract\nThe Chabauty--Kim method is a technique f
or studying the rational points on a curve X using motivic properties of q
uotients U of the fundamental group of X. For specific quotients U\, the m
ethod has been made effective in work of Coleman and later by Balakrishnan
--Dogra\, in the sense that it provides an explicit upper bound on the num
ber 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
applications (joint work with David Corwin\, in progress) towards uniformi
ty results for higher genus curves. A significant part of the proof\, whic
h I will discuss in more detail\, lies in defining a notion of "weight" fo
r Coleman analytic functions\, and showing\, following arguments of Balakr
ishnan--Dogra\, that the number of zeroes of a non-zero Coleman analytic f
unction can be bounded in terms of its weight.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Matschke (Boston University)
DTSTART:20210511T161000Z
DTEND:20210511T163000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/6
DESCRIPTION:Title: A general S-unit equation solver and tables of elliptic curves o
ver number fields\nby Benjamin Matschke (Boston University) as part of
Rational Points and Galois Representations\n\n\nAbstract\nIn this talk we
present work in progress on a new highly optimized\nsolver for general an
d constraint S-unit equations over number fields.\nIt has diophantine appl
ications including asymptotic Fermat theorems\,\nSiegel's method for compu
ting integral points\, and most strikingly for\ncomputing large tables of
elliptic curves over number fields with good\nreduction outside given sets
of primes S. For the latter\, we improved\non the method of Koutsianas (P
arshin\, Shafarevich\, Elkies).\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jackson Morrow (Université de Montréal)
DTSTART:20210511T174000Z
DTEND:20210511T184000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/7
DESCRIPTION:Title: Progress on Mazur's Program B -- a horizontal perspective\nb
y Jackson Morrow (Université de Montréal) as part of Rational Points and
Galois Representations\n\n\nAbstract\nIn this talk\, I will discuss recen
t progress on "Mazur's Program B" --- the problem of classifying all possi
bilities 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 s
trives to classify the composite (non-prime power) images of Galois for an
elliptic curve over $\\mathbb{Q}$. In particular\, I will introduce the n
otion of an entanglement of division fields\, give a group theoretic chara
cterization of an entanglement\, and describe two sets of joint work. The
first is with Harris Daniels where we classify all infinite families of el
liptic 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 prove several results on elliptic curves (and more generally\, principa
lly polarized abelian varieties) over $\\mathbb{Q}$ when the entanglement
occurs over an abelian extension.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbey Bourdon (Wake Forest University)
DTSTART:20210512T140000Z
DTEND:20210512T150000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/8
DESCRIPTION:Title: Families of Sporadic Points on Modular Curves\nby Abbey Bour
don (Wake Forest University) as part of Rational Points and Galois Represe
ntations\n\n\nAbstract\nA closed point $x$ on a curve $C$ is sporadic if t
here are 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 c
orresponds to an elliptic curve with a point of order $N$ defined over a n
umber field of unusually low degree. In this talk\, we will focus on spora
dic points arising from $\\mathbb{Q}$-curves\, which are elliptic curves i
sogenous to their Galois conjugates. In particular\, our investigations ar
e inspired by the following question: Are there only finitely many non-CM
$\\mathbb{Q}$-curves which produce sporadic points on any modular c
urve of the form $X_1(N)$? I will show that an affirmative answer to this
question would imply Serre's Uniformity Conjecture and discuss partial pro
gress in the case of sporadic points of odd degree. This is joint work wit
h Filip Najman.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Sawin (Columbia University)
DTSTART:20210512T153000Z
DTEND:20210512T163000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/9
DESCRIPTION:Title: The Shafarevich conjecture for hypersurfaces in abelian varietie
s\nby Will Sawin (Columbia University) as part of Rational Points and
Galois Representations\n\n\nAbstract\nFaltings proved the statement\, prev
iously conjectured by \nShafarevich\, that there are finitely many abelian
varieties of \ndimension $n$\, defined over a fixed number field\, with g
ood reduction \noutside a fixed finite set of primes\, up to isomorphism.
In joint work \nwith Brian Lawrence\, we prove an analogous finiteness sta
tement for \nhypersurfaces in a fixed abelian variety with good reduction
outside a \nfinite set of primes. I will give an introduction to some of t
he ideas \nin the proof\, which builds on $p$-adic Hodge theory techniques
from work \nof Lawrence and Venkatesh as well as a little-known area of a
lgebraic \ngeometry\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Zureick-Brown (moderator) (Emory University)
DTSTART:20210512T170000Z
DTEND:20210512T180000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/10
DESCRIPTION:Title: Problem discussion session\nby David Zureick-Brown (moderat
or) (Emory University) as part of Rational Points and Galois Representatio
ns\n\n\nAbstract\nThis problem discussion session features advance
contributions (pdf) from \n\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steffen Müller (University of Groningen)
DTSTART:20210511T163000Z
DTEND:20210511T165000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/11
DESCRIPTION:Title: Algorithms for quadratic Chabauty\nby Steffen Müller (Univ
ersity of Groningen) as part of Rational Points and Galois Representations
\n\n\nAbstract\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 poi
nts on certain modular curves. I will discuss a magma-implementation of th
is method and show how to use it in practice. This is joint work with Jenn
ifer Balakrishnan\, Netan Dogra\, Jan Tuitman and Jan Vonk.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Vogt (University of Washington)
DTSTART:20210511T165000Z
DTEND:20210511T171000Z
DTSTAMP:20260404T095118Z
UID:DioGal2021/12
DESCRIPTION:Title: Computing non-surjective primes for abelian surfaces\nby Is
abel Vogt (University of Washington) as part of Rational Points and Galois
Representations\n\n\nAbstract\nLet $A$ be a principally polarized abelian
surface over the rational numbers. Serre proved that there are finitely
many primes ell for which the Galois action on the ell-torsion points of $
A$ is not the entire group of symplectic similitudes $\\mathrm{GSp}_4(\\ma
thbb{F}_\\ell)$. Later\, Dieulefait showed\, conditional on Serre's conje
cture (now a theorem of Khare and Wintenberger)\, that this finite set is
effectively computable. I will report on on-going joint work with Banwait
\, Brumer\, Kim\, Klagsbrun\, Mayle and Srinivasan where we implement this
algorithm and use it to compute nonsurjective primes for all genus 2 curv
es in the LMFDB.\n
LOCATION:https://stable.researchseminars.org/talk/DioGal2021/12/
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