Galois sections and $p$-adic period mappings

TL;DR

This paper proves a finiteness result related to Grothendieck's section conjecture for certain algebraic curves over number fields, extending existing methods and providing insights into $p$-adic period mappings.

Contribution

It establishes the finiteness of the Selmer part of the section set for curves over number fields without CM, confirming a prediction of Grothendieck's section conjecture.

Findings

Finiteness of the Selmer part of the section set for genus ≥2 curves over specific number fields.

Refinement and extension of Lawrence and Venkatesh's method with potential for explicit computations.

Unconditional verification of a prediction of Grothendieck's section conjecture.

Abstract

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\geq2$, 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.