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Seminar

DDC - Diophantine Problems: Uniformity for the Number of Rational Points on a Curve September 21, 2020 (09:00 AM PDT - 10:00 AM PDT)
Parent Program:
Location: MSRI: Online/Virtual
Speaker(s) Philipp Habegger (University of Basel)
Description

This seminar will focus on Diophantine problems in a broad sense, with a view towards (but not limited to) interactions between Number Theory and Logic. Particular attention will be given to topics with the potential of further developments in the context of this MSRI scientific program. This will provide an opportunity for researchers to update on new results, techniques and some of the main problems of the field.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

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To participate in this seminar, please register here: https://www.msri.org/seminars/25206

By Faltings's Theorem, formerly known as the Mordell Conjecture, a smooth projective curve 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 authors, including de Diego, Parshin, Rémond, Vojta, proved upper 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.

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