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Berkeley-Caltech-Stanford Joint Number Theory Seminar

Monday, January 24, 2022
12:30pm to 1:30pm
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The Number of Rational Points on a Curve of Genus > 1
Philipp Habegger, Department of Mathematics and Computer Science, University of Basel,

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 Bombieri, de Diego, Parshin, Rémond, Vojta, proved upper bounds for the number of K-rational points. I will discuss joint work with Vesselin Dimitrov and Ziyang Gao where we prove that the number of points on the curve is bounded from above 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 to the Mordell Conjecture and answer a question of Mazur. I will explain the new feature: an inequality for the Néron-Tate height in a family of abelian varieties. It allows us to bound from above the number of points when the modular height of the curve is sufficiently large.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].