Galois sections and the method of Lawrence-Venkatesh

Grothendieck’s Section Conjecture posits that the set of rational points on a smooth projective curve Y of genus at least two should be equal to a certain “section set” defined purely in terms of the etale fundamental group of Y. In this talk, I will preview some upcoming work with Jakob Stix in which we prove a partial finiteness result for this section set, thereby giving an unconditional verification of a prediction of the Section Conjecture for a general curve Y. We do this by adapting the recent p-adic proof of the Mordell Conjecture due to Brian Lawrence and Akshay Venkatesh.