Lecture series of Prof. Ziyang Gao
Prof. Ziyang Gao
2024-09-19 14:30-16:30
MCM110
Speaker: Prof. Ziyang Gao (Leibniz University Hannover)
Time: 14:30-16:30 September 2 (Monday), September 3 (Tuesday), September 5 (Thursday), September 6 (Friday), September 10 (Tuesday), September 19 (Thursday), 2024
Venue: MCM110
Topic: Uniformity in the number of rational points on curves
Abstract: This mini-course aims to prove the result of Dimitrov-Gao-Habegger on a rather uniform bound on the number of rational points on curves, which depends only on the genus, the degree of the number field, and the Mordell-Weil rank of the Jacobian. We will start by recalling Vojta's proof of the Mordell conjecture, and then prove the height inequality of Gao-Habegger and Dimitrov-Gao-Habegger using the recent work of Yuan-Zhang on adelic line bundles, and finally prove the geometric bigness of the tautological adelic line bundle by studying the generic Betti rank. This last part relies on some functional transcendence result. In the end, we will briefly discuss a more recent work of Gao-Zhang on the Beilinson-Bloch height of the Gross-Schoen cycles.
Title 1: Mordell Conjecture (Part 1 and 2);
Title 2: Zero estimates; preparation for Uniform Mordell-Lang；
Title 3: Adelic line bundles (Part 1 and 2);
Title 4: Adelic line bundle (Part 3); Ax-Schanuel;
Title 5: Betti rank and geometric bigness of the tautological adelic line bundle
Title 6: Some generalization to higher cycles
