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BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART:20200824T160000Z
DTEND:20200824T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/1
DESCRIPTION:Title: Variations on Chabauty\nby Jennifer Balakrishna
n (Boston University) as part of DDC Scientific Program at MSRI - Diophant
ine Problems Seminar\n\n\nAbstract\nWe will describe the Chabauty--Coleman
method and related techniques to determine rational points on curves. In
so doing\, we will highlight some recent examples where these methods have
been used: this includes a problem of Diophantus originally solved by Wet
herell and the problem of the "cursed curve"\, the split Cartan modular cu
rve of level 13.\nThis is joint work with Netan Dogra\, Steffen Mueller\,
Jan Tuitman\, and Jan Vonk.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (CNRS and Université Paris-Sud)
DTSTART:20200831T160000Z
DTEND:20200831T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/2
DESCRIPTION:Title: Some applications of the algebraicity criteria\
nby Yunqing Tang (CNRS and Université Paris-Sud) as part of DDC Scientifi
c Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe classi
cal Borel--Dwork rationality criterion provides a sufficient condition for
a formal power series of rational coefficients to be (the Taylor expansio
n of) a rational function in terms of its radii of convergence (in some qu
otient representation) at all places. There are various generalizations of
this criterion\; in particular\, a special case of the Grothendieck--Katz
p-curvature conjecture is proved by Chudnovsky--Chudnovsky\, André\, and
Bost using their algebraicity criteria\, which are generalizations of the
Borel--Dwork criterion. In this talk\, I will recall the p-curvature conj
ecture and these algebraicity criteria and then I will discuss some other
applications of these criteria. Part of the talk is based on the joint wor
k in progress with Frank Calegari and Vesselin Dimitrov on p-adic zeta val
ues.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART:20200914T160000Z
DTEND:20200914T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/3
DESCRIPTION:Title: Goldfeld's conjecture and congruences between Heegn
er points\nby Chao Li (Columbia University) as part of DDC Scientific
Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nGiven an ell
iptic curve $E$ over $\\mathbb{Q}$\, a celebrated conjecture of Goldfeld a
sserts that a positive proportion of its quadratic twists should have anal
ytic rank $0$ (resp. $1$). We show this conjecture holds whenever $E$ has
a rational $3$-isogeny. We also prove the analogous result for the sextic
twists family. For a more general elliptic curve $E$\, we show that the nu
mber of quadratic twists of $E$ up to twisting discriminant $X$ of analyti
c rank $0$ (resp. $1$) is $>> X/log^{5/6}X$\, improving the current best g
eneral bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Pere
lli--Pomykala). We prove these results by establishing a congruence formul
a between p-adic logarithms of Heegner points. This is joint work with Dan
iel Kriz.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Habegger (University of Basel)
DTSTART:20200921T160000Z
DTEND:20200921T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/4
DESCRIPTION:Title: Uniformity for the Number of Rational Points on a C
urve\nby Philipp Habegger (University of Basel) as part of DDC Scienti
fic Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nBy Falti
ngs's Theorem\, formerly known as the Mordell Conjecture\, a smooth projec
tive curve of genus at least 2 that is defined over a number field K has a
t most finitely many K-rational points. Votja later gave a second proof. M
any authors\, including de Diego\, Parshin\, Rémond\, Vojta\, proved uppe
r bounds for the number of K-rational points. In this talk I will discuss
joint work with Vesselin Dimitrov and Ziyang Gao. We show that the number
of points on the curve is bounded as a function of K\, the genus\, and the
rank of the Mordell-Weil group of the curve's Jacobian. We follow Vojta's
approach and complement it by bounding the number of "small points" using
a new lower bound for the Néron-Tate height.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (Massachusetts Institute of Technology)
DTSTART:20200928T160000Z
DTEND:20200928T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/5
DESCRIPTION:Title: $2^k$-Selmer groups\, the Cassels-Tate pairing\, an
d Goldfeld's conjecture\nby Alexander Smith (Massachusetts Institute o
f Technology) as part of DDC Scientific Program at MSRI - Diophantine Prob
lems Seminar\n\n\nAbstract\nTake $E$ to be an elliptic curve over a number
field whose four torsion obeys certain technical conditions. In this talk
\, we will outline a proof that $100\\%$ of the quadratic twists of $E$ ha
ve rank at most one. To do this\, we will find the distribution of $2^k$-S
elmer ranks in this family for every $k > 1$.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neer Bhardwaj (University of Illinois at Urbana-Champaign)
DTSTART:20201005T160000Z
DTEND:20201005T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/6
DESCRIPTION:Title: On the Pila-Wilkie Theorem\nby Neer Bhardwaj (U
niversity of Illinois at Urbana-Champaign) as part of DDC Scientific Progr
am at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nWe prove Pila and
Wilkie’s Counting theorem\, following the original paper\, but exploit
cell decomposition more thoroughly to simplify the deduction from its main
ingredients. Our approach in particular completely avoids ‘regular’ o
r C^1 smooth points\, and related technology\; which also allows simplific
ations around Pila’s ‘block family’ refinement of the result.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arno Fehm (Technische Universität Dresden)
DTSTART:20201019T160000Z
DTEND:20201019T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/7
DESCRIPTION:Title: Diophantine problems over large fields\nby Arno
Fehm (Technische Universität Dresden) as part of DDC Scientific Program
at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nA field K is large i
f every smooth K-curve with a K-rational point has\ninfinitely many of the
se. Large fields were introduced in the context of\nGalois theory\, where
they now play an important role\, but they happen to\nshow up naturally al
so in several other areas\, such as valuation theory\,\narithmetic geometr
y and model theory. In this talk I will give a brief\nintroduction to larg
e fields\, will survey some results regarding\ndiophantine sets involving
large fields\, and will then explain in more\ndetail why over a large fiel
d one usually cannot find an abelian variety\nof finite Mordell-Weil rank\
, a fact (obtained in joint work with S.\nPetersen) that is relevant in th
e context of Hilbert's tenth problem.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Eterovic (UC Berkeley)
DTSTART:20201026T160000Z
DTEND:20201026T170000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/8
DESCRIPTION:Title: The Existential Closedness Problem for the Modular
$j$-function\nby Sebastian Eterovic (UC Berkeley) as part of DDC Scien
tific Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe ex
istential closedness problem for $j$ asks to find a "minimal" set of geome
tric conditions that an algebraic variety $V\\subset\\mathbb{C}^{2n}$ shou
ld satisfy in order to ensure that it has a point of the form $(z_1\,\\ldo
ts\,z_n\,j(z_1)\,\\ldots\,j(z_n))$. Furthermore\, one wants to know if for
every finitely generated field $F$ there is a generic point in $V$ over $
F$ of this form. In this talk I will introduce the problem\, I will presen
t some of the known results\, and I will explain how it relates to some ve
ry important open conjectures such as the Zilber-Pink conjecture and the m
odular Schanuel conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicholas Triantafillou (University of Georgia)
DTSTART:20201102T170000Z
DTEND:20201102T180000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/9
DESCRIPTION:Title: Nonexistence of exceptional units via Skolem-Chabau
ty's method\nby Nicholas Triantafillou (University of Georgia) as part
of DDC Scientific Program at MSRI - Diophantine Problems Seminar\n\n\nAbs
tract\nAn exceptional (S-)unit is a unit x in a ring of (S-)integers of a
number field K such that 1-x is also an (S-)unit. For fixed K and S\, the
set of exceptional S-units is finite by work of Siegel from the early 1900
s. In the hundred years since\, exceptional S-units have found wide-rangin
g applications\, including to enumerating elliptic curves with good reduct
ion outside a fixed set of primes and to proving "asymptotic" versions of
Fermat's last theorem.\n\nIn this talk\, we give an elementary p-adic proo
f of a new nonexistence result on exceptional units: there are no exceptio
nal units in number fields of degree prime to 3 where 3 splits completely.
We will also explain the geometric inspiration for the proof -- a version
of Skolem-Chabauty's method for finding integral points on curves. Time p
ermitting\, we will discuss an application to periodic points of odd order
in arithmetic dynamics.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Ingram (York University)
DTSTART:20201109T170000Z
DTEND:20201109T180000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/10
DESCRIPTION:Title: The critical height of an endomorphism of projecti
ve space\nby Patrick Ingram (York University) as part of DDC Scientifi
c Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe critic
al height of an endomorphism of projective space is a candidate for a “c
anonical” height on the corresponding moduli space of dynamical systems.
I will survey some results on the critical height\, and mention a few ope
n problems.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Goettingen University)
DTSTART:20201116T170000Z
DTEND:20201116T180000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/11
DESCRIPTION:Title: Counting rational points on and close to algebraic
varieties\nby Damaris Schindler (Goettingen University) as part of DD
C Scientific Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\
nGiven an algebraic variety over the rational numbers\, how can we decide
if we can find rational/integral points on it? And assuming that there are
rational/integral points\, under what circumstances can we count them in
a meaningful way? What can we say about the number of rational points clos
e to algebraic varieties or smooth manifolds? These and related questions
are going to be the topic of this talk. We are going to focus on situation
s where analytic tools play a key role in finding answers.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Pop (University of Pennsylvania)
DTSTART:20201123T170000Z
DTEND:20201123T180000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/12
DESCRIPTION:Title: Generalizations of CCT (II)\nby Florian Pop (U
niversity of Pennsylvania) as part of DDC Scientific Program at MSRI - Dio
phantine Problems Seminar\n\n\nAbstract\nThe CCT (Colliot-Thelene Conjectu
re) over a number field k is about giving birational conditions on morphis
ms of proper smooth k-varieties which imply surjectivity on the local rati
onal points for almost all localization of k.\nThe CCT was proved in a str
onger form by Denef (2017)\, and Loughran-Skorobogatov-Smeets (2019) gave
necessary and sufficient conditions for Denef's result to hold. This talk
(in some sense a follow-up to my 2019 Fields Institute talk) is about furt
her generalizations of the afore mentioned results in several ways\, by re
laxing both the hypothesis on the bases field k\, and the conditions on th
e varieties involved (e.g. no properness or smoothness\, etc.). The point
in my approach is to employ special forms of the AKE (whereas the CCT was
—among other things—aimed at giving an arithmetic geometry proof of th
e AKE). I will also mention a few open questions and potential research di
rections.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpoge (Columbia University)
DTSTART:20201130T170000Z
DTEND:20201130T180000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/13
DESCRIPTION:Title: On Hilbert's tenth problem in two variables\nb
y Levent Alpoge (Columbia University) as part of DDC Scientific Program at
MSRI - Diophantine Problems Seminar\n\n\nAbstract\nIn joint work with Bri
an Lawrence\, we show that\, assuming standard\nmotivic conjectures (Fonta
ine-Mazur\, Grothendieck-Serre\, Hodge\, Tate)\,\nthere is a finite-time a
lgorithm that\, on input $(K\,C)$ with $K$ a number\nfield and $C/K$ a smo
oth projective hyperbolic (i.e. genus $> 1$) curve\,\noutputs $C(K)$. The
algorithm has the property that\, if it terminates\,\nthe output is uncond
itionally correct --- one uses the conjectures to\nshow that it always ter
minates in finite time.\n\nOn the other hand\, in certain cases (i.e. afte
r imposing conditions on\n$K$ and $C$) there is an unconditional finite-ti
me algorithm to compute\n$(K\,C)\\mapsto C(K)$\, using potential modularit
y theorems instead.\nExample: given $K$ totally real of odd degree and $a\
\in K^\\times$\, one can\neffectively compute $C_a(K)$ where $C_a : x^6 +
4y^3 = a^2$.\n\nI will focus on the first of these two results but will tr
y to mention\nat least the ideas that go into the second if time permits.\
n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Umberto Zannier (Scuola Normale Superiore)
DTSTART:20201207T170000Z
DTEND:20201207T180000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/14
DESCRIPTION:Title: Torsion values of sections\, elliptical billiards
and diophantine problems in dynamics\nby Umberto Zannier (Scuola Norma
le Superiore) as part of DDC Scientific Program at MSRI - Diophantine Prob
lems Seminar\n\n\nAbstract\nWe shall consider sections of (products of) el
liptic schemes\,\nand their "torsion values". For instance\, what can be s
aid\nof the complex numbers $b$ for which $(2\, \\sqrt{2(2-b)})$ is torsio
n\non $y^2=x(x-1)(x-b)$?\nIn particular\, we shall recall results of "Mani
n-Mumford type"\nand illustrate some applications to elliptical billiards.
\nFinally\, we shall frame these issues as special cases of\na general que
stion in arithmetic dynamics\, which can be\ntreated with different method
s\, depending on the context.\n(Most results refer to work with Pietro Cor
vaja and David Masser.)\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Sarnak (Princeton University and IAS)
DTSTART:20201214T170000Z
DTEND:20201214T180000Z
DTSTAMP:20260404T094121Z
UID:DiophantineProblemsMSRI/15
DESCRIPTION:Title: Applications of points on subvarieties of tori
\nby Peter Sarnak (Princeton University and IAS) as part of DDC Scientific
Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe interse
ction of the division group of a finitely generated subgroup of a torus wi
th an algebraic sub-variety has been understood for some time (Lang\, Laur
ent ..). After a brief review of some of the tools in the analysis and the
ir recent extensions (André-Oort conjectures)\, we give some old and new
applications\; in particular to the additive structure of the spectra of m
etric graphs and crystalline measures.\nThe last is joint work with P. Kur
asov.\n
LOCATION:https://stable.researchseminars.org/talk/DiophantineProblemsMSRI/
15/
END:VEVENT
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