A Faltings--Mordell Theorem for Selmer Sections

In 1983, shortly after Faltings’ resolution of the Mordell Conjecture, Grothendieck formulated his famous Section Conjecture, positing that the set of rational points on a projective curve Y of genus at least two should be equal to a certain section set defined in terms of the etale fundamental group of Y. To this day, this conjecture remains wide open, with only a small handful of very special examples known. In this talk, I will discuss recent work with Jakob Stix, in which we proved a Mordell-like finiteness theorem for the "Selmer" part of the section set for any smooth projective curve Y of genus at least 2 over the rationals. This generalises the Faltings--Mordell Theorem, and implies strong constraints on the finite descent locus from obstruction theory. The key new idea in our proof is an adaptation of the recent proof of Mordell by Lawrence and Venkatesh to the study of the Selmer section set.