This talk concerns two generalisations of the Chabauty method, which studies the rational points on a curve X by embedding it inside its Jacobian. The first is Kim's non-abelian Chabauty programme, which replaces the Jacobian with a sequence of "Selmer schemes" produced from the fundamental group of X. The second is the geometric quadratic Chabauty method of Edixhoven and Lido, which replaces the Jacobian instead by a Gm-torsor over the Jacobian. These two generalisations are related by work of Hashimoto, Duque Rosero and Spelier.

The main aim of this talk is to explain why these two approaches are related, with the punchline being that torsors under tori over abelian varieties have fundamental groups which exactly realise the quotients of pi_1(X) studied in the quadratic part of Chabauty—Kim. Time permitting, we will also explain how this perspective connects quadratic Chabauty with a new kind of unlikely intersections problem, and what this tells us about the structure of the quadratic Chabauty locus.