Galois sections and $p$-adic period mappings

Abstract

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\ge 2$, we prove that the image of the "Selmer" part of Grothendieck's section set inside the $K_v$-rational points $Y(K_v)$ is finite for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck's section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations.

59 pages, comments welcome