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arXiv · 2504.04484

Ramified periods and field of definition

Giuseppe Ancona·Dragoş Frăţilă·Alberto Vezzani
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Abstract

Let $L/K$ be an extension of number fields that is ramified above $p$. We give a new obstruction to the descent to $K$ of smooth projective varieties defined over $L$. The obstruction is a matrix of $p$-adic numbers that we call ``ramified periods'' arising from the comparison isomorphism between de Rham cohomology and crystalline cohomology. As an application, we give simple examples of hyperelliptic curves over $\mathbb{Q}(\sqrt p)$ that are isomorphic to their Galois conjugates but such that their Jacobians do not descend to $\mathbb{Q}$ even up to isogeny.

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Giuseppe Ancona, Dragoş Frăţilă, Alberto Vezzani. 2025-04-06. Ramified periods and field of definition. https://arxiv.org/abs/2504.04484

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