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arXiv · 2204.13674

Galois sections and $p$-adic period mappings

L. Alexander Betts·Jakob Stix
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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.

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L. Alexander Betts, Jakob Stix. 2022-04-28. Galois sections and $p$-adic period mappings. https://arxiv.org/abs/2204.13674

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