Geometric quadratic Chabauty (Colloquium AGA)

Determining all rational points on a curve of genus at least $2$ can be difficult. Chabauty's method (1941) is to intersect, for a prime number $p$, in the $p$-adic Lie group of $p$-adic points of the jacobian, the closure of the Mordell-Weil group with the $p$-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed.

 

Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$).

 

This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only "simple algebraic geometry" (line bundles over the jacobian and models over the integers).