Seminar

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

Abstract: For a (projective or affine) hyperbolic curve C over a number field, the non-abelian Chabauty's method of Minhyong Kim defines a sequence of nested subsets (depending on a positive integer n) of the p-adic points of C containing the rational or integral points of C. The case n=1 is Chabauty's method, and Kim shows certain well-known conjectures on Galois representations imply these sets are finite for sufficiently large n. Therefore, computing these sets could lead to an approach to Effective Faltings' Theorem, the problem of enumerating rational or integral points on hyperbolic curves.

The most successful method to-date for computing these sets is known as Quadratic Chabauty, and it applies when n=2. An algorithm for Effective Faltings in general would likely require being able to compute for arbitrary n. Previous work of Dan-Cohen--Wewers and C--Dan-Cohen has computed some cases with n=4 and C a genus 0 curve using a Tannakian motivic formalism. I will report on ongoing work to extend these ideas to higher genus, including an example in genus 1 with n=3.

Beyond Quadratic Chabauty