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SUMMARY:Jennifer Balakrishnan (Boston University)
DTSTART;VALUE=DATE-TIME:20200824T160000Z
DTEND;VALUE=DATE-TIME:20200824T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
UID:DiophantineProblemsMSRI/1
DESCRIPTION:Title: Variations on Chabauty\nby Jennifer Balakrishnan (Bosto
n University) as part of DDC Scientific Program at MSRI - Diophantine Prob
lems 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 us
ed: this includes a problem of Diophantus originally solved by Wetherell a
nd the problem of the "cursed curve"\, the split Cartan modular curve of l
evel 13.\nThis is joint work with Netan Dogra\, Steffen Mueller\, Jan Tuit
man\, and Jan Vonk.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunqing Tang (CNRS and Université Paris-Sud)
DTSTART;VALUE=DATE-TIME:20200831T160000Z
DTEND;VALUE=DATE-TIME:20200831T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
UID:DiophantineProblemsMSRI/2
DESCRIPTION:Title: Some applications of the algebraicity criteria\nby Yunq
ing Tang (CNRS and Université Paris-Sud) as part of DDC Scientific Progra
m at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe classical Bore
l--Dwork rationality criterion provides a sufficient condition for a forma
l power series of rational coefficients to be (the Taylor expansion of) a
rational function in terms of its radii of convergence (in some quotient r
epresentation) at all places. There are various generalizations of this cr
iterion\; in particular\, a special case of the Grothendieck--Katz p-curva
ture conjecture is proved by Chudnovsky--Chudnovsky\, André\, and Bost us
ing their algebraicity criteria\, which are generalizations of the Borel--
Dwork criterion. In this talk\, I will recall the p-curvature conjecture a
nd these algebraicity criteria and then I will discuss some other applicat
ions of these criteria. Part of the talk is based on the joint work in pro
gress with Frank Calegari and Vesselin Dimitrov on p-adic zeta values.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chao Li (Columbia University)
DTSTART;VALUE=DATE-TIME:20200914T160000Z
DTEND;VALUE=DATE-TIME:20200914T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
UID:DiophantineProblemsMSRI/3
DESCRIPTION:Title: Goldfeld's conjecture and congruences between Heegner p
oints\nby Chao Li (Columbia University) as part of DDC Scientific Program
at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nGiven an elliptic cu
rve $E$ over $\\mathbb{Q}$\, a celebrated conjecture of Goldfeld asserts t
hat a positive proportion of its quadratic twists should have analytic ran
k $0$ (resp. $1$). We show this conjecture holds whenever $E$ has a ration
al $3$-isogeny. We also prove the analogous result for the sextic twists f
amily. 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 b
ound towards Goldfeld's conjecture due to Ono--Skinner (resp. Perelli--Pom
ykala). We prove these results by establishing a congruence formula betwee
n p-adic logarithms of Heegner points. This is joint work with Daniel Kriz
.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Habegger (University of Basel)
DTSTART;VALUE=DATE-TIME:20200921T160000Z
DTEND;VALUE=DATE-TIME:20200921T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
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 Prog
ram at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nBy Faltings's Th
eorem\, formerly known as the Mordell Conjecture\, a smooth projective cur
ve of genus at least 2 that is defined over a number field K has at most f
initely many K-rational points. Votja later gave a second proof. Many auth
ors\, including de Diego\, Parshin\, Rémond\, Vojta\, proved upper bounds
for the number of K-rational points. In this talk I will discuss joint wo
rk with Vesselin Dimitrov and Ziyang Gao. We show that the number of point
s 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 approac
h and complement it by bounding the number of "small points" using a new l
ower bound for the Néron-Tate height.\n
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BEGIN:VEVENT
SUMMARY:Alexander Smith (Massachusetts Institute of Technology)
DTSTART;VALUE=DATE-TIME:20200928T160000Z
DTEND;VALUE=DATE-TIME:20200928T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
UID:DiophantineProblemsMSRI/5
DESCRIPTION:Title: $2^k$-Selmer groups\, the Cassels-Tate pairing\, and Go
ldfeld's conjecture\nby Alexander Smith (Massachusetts Institute of Techno
logy) as part of DDC Scientific Program at MSRI - Diophantine Problems Sem
inar\n\n\nAbstract\nTake $E$ to be an elliptic curve over a number field w
hose four torsion obeys certain technical conditions. In this talk\, we wi
ll 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 ra
nks in this family for every $k > 1$.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neer Bhardwaj (University of Illinois at Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20201005T160000Z
DTEND;VALUE=DATE-TIME:20201005T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
UID:DiophantineProblemsMSRI/6
DESCRIPTION:Title: On the Pila-Wilkie Theorem\nby Neer Bhardwaj (Universit
y of Illinois at Urbana-Champaign) as part of DDC Scientific Program at MS
RI - 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
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arno Fehm (Technische Universität Dresden)
DTSTART;VALUE=DATE-TIME:20201019T160000Z
DTEND;VALUE=DATE-TIME:20201019T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
UID:DiophantineProblemsMSRI/7
DESCRIPTION:Title: Diophantine problems over large fields\nby Arno Fehm (T
echnische 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. Larg
e fields were introduced in the context of\nGalois theory\, where they now
play an important role\, but they happen to\nshow up naturally also in se
veral other areas\, such as valuation theory\,\narithmetic geometry and mo
del theory. In this talk I will give a brief\nintroduction to large fields
\, will survey some results regarding\ndiophantine sets involving large fi
elds\, and will then explain in more\ndetail why over a large field one us
ually cannot find an abelian variety\nof finite Mordell-Weil rank\, a fact
(obtained in joint work with S.\nPetersen) that is relevant in the contex
t of Hilbert's tenth problem.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Eterovic (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20201026T160000Z
DTEND;VALUE=DATE-TIME:20201026T170000Z
DTSTAMP;VALUE=DATE-TIME:20201029T102956Z
UID:DiophantineProblemsMSRI/8
DESCRIPTION:Title: The Existential Closedness Problem for the Modular $j$-
function\nby Sebastian Eterovic (UC Berkeley) as part of DDC Scientific Pr
ogram at MSRI - Diophantine Problems Seminar\n\n\nAbstract\nThe existentia
l closedness problem for $j$ asks to find a "minimal" set of geometric con
ditions that an algebraic variety $V\\subset\\mathbb{C}^{2n}$ should satis
fy 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 f
initely generated field $F$ there is a generic point in $V$ over $F$ of th
is form. In this talk I will introduce the problem\, I will present some o
f the known results\, and I will explain how it relates to some very impor
tant open conjectures such as the Zilber-Pink conjecture and the modular S
chanuel conjecture.\n
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