Diophantine Geometry and Grothendieck's Section Set

In 1983, Alexander Grothendieck proposed his anabelian programme: the study of rational points on hyperbolic curves by means of their fundamental groups. The centrepiece of this programme was his Section Conjecture, which posited that the set of rational points should be equal to a certain section set defined purely in terms of fundamental groups. Although the Section Conjecture remains wide open, the ideas underlying it have proven remarkably effective in the study of rational points. For example, Kim's method of non-abelian Chabauty uses unipotent fundamental groups to give a "computable" obstruction to rational points, sufficient to compute them in practice in many cases.

 

In this talk, I will explain how -- in a reversal of the usual flow of ideas -- one can use techniques from Diophantine geometry to study Grothendieck's programme and gain concrete understanding about the mysterious section set. In particular, one can prove Mordell-like finiteness theorems for the section set, as well as compute it in (so far) one particular case. This encompasses joint works with Jakob Stix, and with Theresa Kumpitsch and Martin Lüdtke.