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BEGIN:VEVENT
SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART:20200824T160000Z
DTEND:20200824T170000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/1
DESCRIPTION:Title: Variations on Chabauty\nby Jennifer Balakrishnan (Bost
on University) as part of DDC Scientific Program at MSRI - Diophantine Pro
blems Seminar\n\n\nAbstract\nWe will describe the Chabauty--Coleman method
and related techniques to determine rational points on curves. In so doin
g\, we will highlight some recent examples where these methods have been u
sed: this includes a problem of Diophantus originally solved by Wetherell
and the problem of the "cursed curve"\, the split Cartan modular curve of
level 13.\nThis is joint work with Netan Dogra\, Steffen Mueller\, Jan Tui
tman\, and Jan Vonk.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (CNRS and Université Paris-Sud)
DTSTART:20200831T160000Z
DTEND:20200831T170000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/2
DESCRIPTION:Title: Some applications of the algebraicity criteria\nby Yun
qing Tang (CNRS and Université Paris-Sud) as part of DDC Scientific Progr
am at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe classical Bor
el--Dwork rationality criterion provides a sufficient condition for a form
al power series of rational coefficients to be (the Taylor expansion of) a
rational function in terms of its radii of convergence (in some quotient
representation) at all places. There are various generalizations of this c
riterion\; in particular\, a special case of the Grothendieck--Katz p-curv
ature conjecture is proved by Chudnovsky--Chudnovsky\, André\, and Bost u
sing their algebraicity criteria\, which are generalizations of the Borel-
-Dwork criterion. In this talk\, I will recall the p-curvature conjecture
and these algebraicity criteria and then I will discuss some other applica
tions of these criteria. Part of the talk is based on the joint work in pr
ogress with Frank Calegari and Vesselin Dimitrov on p-adic zeta values.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART:20200914T160000Z
DTEND:20200914T170000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/3
DESCRIPTION:Title: Goldfeld's conjecture and congruences between Heegner poin
ts\nby Chao Li (Columbia University) as part of DDC Scientific Program
at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nGiven an elliptic c
urve $E$ over $\\mathbb{Q}$\, a celebrated conjecture of Goldfeld asserts
that a positive proportion of its quadratic twists should have analytic ra
nk $0$ (resp. $1$). We show this conjecture holds whenever $E$ has a ratio
nal $3$-isogeny. We also prove the analogous result for the sextic twists
family. For a more general elliptic curve $E$\, we show that the number of
quadratic twists of $E$ up to twisting discriminant $X$ of analytic rank
$0$ (resp. $1$) is $>> X/log^{5/6}X$\, improving the current best general
bound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Po
mykala). We prove these results by establishing a congruence formula betwe
en p-adic logarithms of Heegner points. This is joint work with Daniel Kri
z.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Habegger (University of Basel)
DTSTART:20200921T160000Z
DTEND:20200921T170000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/4
DESCRIPTION:Title: Uniformity for the Number of Rational Points on a Curve\nby Philipp Habegger (University of Basel) as part of DDC Scientific Pro
gram at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nBy Faltings's T
heorem\, formerly known as the Mordell Conjecture\, a smooth projective cu
rve of genus at least 2 that is defined over a number field K has at most
finitely many K-rational points. Votja later gave a second proof. Many aut
hors\, including de Diego\, Parshin\, Rémond\, Vojta\, proved upper bound
s for the number of K-rational points. In this talk I will discuss joint w
ork with Vesselin Dimitrov and Ziyang Gao. We show that the number of poin
ts on the curve is bounded as a function of K\, the genus\, and the rank o
f the Mordell-Weil group of the curve's Jacobian. We follow Vojta's approa
ch and complement it by bounding the number of "small points" using a new
lower bound for the Néron-Tate height.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Smith (Massachusetts Institute of Technology)
DTSTART:20200928T160000Z
DTEND:20200928T170000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/5
DESCRIPTION:Title: $2^k$-Selmer groups\, the Cassels-Tate pairing\, and Goldf
eld's conjecture\nby Alexander Smith (Massachusetts Institute of Techn
ology) as part of DDC Scientific Program at MSRI - Diophantine Problems Se
minar\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 w
ill outline a proof that $100\\%$ of the quadratic twists of $E$ have rank
at most one. To do this\, we will find the distribution of $2^k$-Selmer r
anks in this family for every $k > 1$.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neer Bhardwaj (University of Illinois at Urbana-Champaign)
DTSTART:20201005T160000Z
DTEND:20201005T170000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/6
DESCRIPTION:Title: On the Pila-Wilkie Theorem\nby Neer Bhardwaj (Universi
ty of Illinois at Urbana-Champaign) as part of DDC Scientific Program at M
SRI - Diophantine Problems Seminar\n\n\nAbstract\nWe prove Pila and Wilkie
’s Counting theorem\, following the original paper\, but exploit cell de
composition more thoroughly to simplify the deduction from its main ingred
ients. Our approach in particular completely avoids ‘regular’ or C^1 s
mooth points\, and related technology\; which also allows simplifications
around Pila’s ‘block family’ refinement of the result.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arno Fehm (Technische Universität Dresden)
DTSTART:20201019T160000Z
DTEND:20201019T170000Z
DTSTAMP:20260422T212556Z
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 if every
smooth K-curve with a K-rational point has\ninfinitely many of these. Lar
ge fields were introduced in the context of\nGalois theory\, where they no
w play an important role\, but they happen to\nshow up naturally also in s
everal other areas\, such as valuation theory\,\narithmetic geometry and m
odel theory. In this talk I will give a brief\nintroduction to large field
s\, will survey some results regarding\ndiophantine sets involving large f
ields\, and will then explain in more\ndetail why over a large field one u
sually cannot find an abelian variety\nof finite Mordell-Weil rank\, a fac
t (obtained in joint work with S.\nPetersen) that is relevant in the conte
xt of Hilbert's tenth problem.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Eterovic (UC Berkeley)
DTSTART:20201026T160000Z
DTEND:20201026T170000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/8
DESCRIPTION:Title: The Existential Closedness Problem for the Modular $j$-fun
ction\nby Sebastian Eterovic (UC Berkeley) as part of DDC Scientific P
rogram at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe existenti
al closedness problem for $j$ asks to find a "minimal" set of geometric co
nditions that an algebraic variety $V\\subset\\mathbb{C}^{2n}$ should sati
sfy in order to ensure that it has a point of the form $(z_1\,\\ldots\,z_n
\,j(z_1)\,\\ldots\,j(z_n))$. Furthermore\, one wants to know if for every
finitely generated field $F$ there is a generic point in $V$ over $F$ of t
his form. In this talk I will introduce the problem\, I will present some
of the known results\, and I will explain how it relates to some very impo
rtant open conjectures such as the Zilber-Pink conjecture and the modular
Schanuel conjecture.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicholas Triantafillou (University of Georgia)
DTSTART:20201102T170000Z
DTEND:20201102T180000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/9
DESCRIPTION:Title: Nonexistence of exceptional units via Skolem-Chabauty's me
thod\nby Nicholas Triantafillou (University of Georgia) as part of DDC
Scientific Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\n
An 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 1900s. In t
he hundred years since\, exceptional S-units have found wide-ranging appli
cations\, including to enumerating elliptic curves with good reduction out
side 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 proof of a
new nonexistence result on exceptional units: there are no exceptional uni
ts in number fields of degree prime to 3 where 3 splits completely. We wil
l also explain the geometric inspiration for the proof -- a version of Sko
lem-Chabauty's method for finding integral points on curves. Time permitti
ng\, we will discuss an application to periodic points of odd order in ari
thmetic dynamics.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Ingram (York University)
DTSTART:20201109T170000Z
DTEND:20201109T180000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/10
DESCRIPTION:Title: The critical height of an endomorphism of projective spac
e\nby Patrick Ingram (York University) as part of DDC Scientific Progr
am at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe critical heig
ht of an endomorphism of projective space is a candidate for a “canonica
l” height on the corresponding moduli space of dynamical systems. I will
survey some results on the critical height\, and mention a few open probl
ems.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Goettingen University)
DTSTART:20201116T170000Z
DTEND:20201116T180000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/11
DESCRIPTION:Title: Counting rational points on and close to algebraic variet
ies\nby Damaris Schindler (Goettingen University) as part of DDC Scien
tific Program at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nGiven
an algebraic variety over the rational numbers\, how can we decide if we c
an find rational/integral points on it? And assuming that there are ration
al/integral points\, under what circumstances can we count them in a meani
ngful way? What can we say about the number of rational points close to al
gebraic varieties or smooth manifolds? These and related questions are goi
ng to be the topic of this talk. We are going to focus on situations where
analytic tools play a key role in finding answers.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Pop (University of Pennsylvania)
DTSTART:20201123T170000Z
DTEND:20201123T180000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/12
DESCRIPTION:Title: Generalizations of CCT (II)\nby Florian Pop (Universi
ty of Pennsylvania) as part of DDC Scientific Program at MSRI - Diophantin
e Problems Seminar\n\n\nAbstract\nThe CCT (Colliot-Thelene Conjecture) ove
r a number field k is about giving birational conditions on morphisms of p
roper smooth k-varieties which imply surjectivity on the local rational po
ints for almost all localization of k.\nThe CCT was proved in a stronger f
orm by Denef (2017)\, and Loughran-Skorobogatov-Smeets (2019) gave necessa
ry and sufficient conditions for Denef's result to hold. This talk (in som
e sense a follow-up to my 2019 Fields Institute talk) is about further gen
eralizations of the afore mentioned results in several ways\, by relaxing
both the hypothesis on the bases field k\, and the conditions on the varie
ties involved (e.g. no properness or smoothness\, etc.). The point in my a
pproach is to employ special forms of the AKE (whereas the CCT was—among
other things—aimed at giving an arithmetic geometry proof of the AKE).
I will also mention a few open questions and potential research directions
.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Levent Alpoge (Columbia University)
DTSTART:20201130T170000Z
DTEND:20201130T180000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/13
DESCRIPTION:Title: On Hilbert's tenth problem in two variables\nby Leven
t Alpoge (Columbia University) as part of DDC Scientific Program at MSRI -
Diophantine Problems Seminar\n\n\nAbstract\nIn joint work with Brian Lawr
ence\, we show that\, assuming standard\nmotivic conjectures (Fontaine-Maz
ur\, Grothendieck-Serre\, Hodge\, Tate)\,\nthere is a finite-time algorith
m that\, on input $(K\,C)$ with $K$ a number\nfield and $C/K$ a smooth pro
jective hyperbolic (i.e. genus $> 1$) curve\,\noutputs $C(K)$. The algorit
hm has the property that\, if it terminates\,\nthe output is unconditional
ly correct --- one uses the conjectures to\nshow that it always terminates
in finite time.\n\nOn the other hand\, in certain cases (i.e. after impos
ing conditions on\n$K$ and $C$) there is an unconditional finite-time algo
rithm to compute\n$(K\,C)\\mapsto C(K)$\, using potential modularity theor
ems 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 try to me
ntion\nat least the ideas that go into the second if time permits.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Umberto Zannier (Scuola Normale Superiore)
DTSTART:20201207T170000Z
DTEND:20201207T180000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/14
DESCRIPTION:Title: Torsion values of sections\, elliptical billiards and dio
phantine problems in dynamics\nby Umberto Zannier (Scuola Normale Supe
riore) as part of DDC Scientific Program at MSRI - Diophantine Problems Se
minar\n\n\nAbstract\nWe shall consider sections of (products of) elliptic
schemes\,\nand their "torsion values". For instance\, what can be said\nof
the complex numbers $b$ for which $(2\, \\sqrt{2(2-b)})$ is torsion\non $
y^2=x(x-1)(x-b)$?\nIn particular\, we shall recall results of "Manin-Mumfo
rd type"\nand illustrate some applications to elliptical billiards.\nFinal
ly\, we shall frame these issues as special cases of\na general question i
n arithmetic dynamics\, which can be\ntreated with different methods\, dep
ending on the context.\n(Most results refer to work with Pietro Corvaja an
d David Masser.)\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Sarnak (Princeton University and IAS)
DTSTART:20201214T170000Z
DTEND:20201214T180000Z
DTSTAMP:20260422T212556Z
UID:DiophantineProblemsMSRI/15
DESCRIPTION:Title: Applications of points on subvarieties of tori\nby Pe
ter Sarnak (Princeton University and IAS) as part of DDC Scientific Progra
m at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe intersection o
f the division group of a finitely generated subgroup of a torus with an a
lgebraic sub-variety has been understood for some time (Lang\, Laurent ..)
. After a brief review of some of the tools in the analysis and their rece
nt extensions (André-Oort conjectures)\, we give some old and new applica
tions\; in particular to the additive structure of the spectra of metric g
raphs and crystalline measures.\nThe last is joint work with P. Kurasov.\n
LOCATION:https://researchseminars.org/talk/DiophantineProblemsMSRI/15/
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