Ramified periods and field of definition

TL;DR

The paper addresses descent of smooth projective varieties from a ramified extension $L/K$ at a prime $p$, introducing ramified periods from the de Rham–crystalline comparison to obstruct descent. It develops a refined framework—ramified periods and filtered ramified periods—within the Hyodo–Kato formalism to detect descent obstructions, and defines a minimal-period invariant that must vanish if a model over $K$ exists. Applying this to explicit hyperelliptic curves defined by $\mathcal{C}_a: y^2=x^{2g+2}-a$ with $a\in\mathbb{Q}(\sqrt p)^{\times}$, the authors produce infinitely many examples where $\mathcal{C}_a$ is isomorphic to its Galois conjugate but $J(\mathcal{C}_a)$ does not descend to $\mathbb{Q}$ up to isogeny, using Pell-type data and a nontrivial minimal-period computation. The results illuminate how ramification in $p$-adic Hodge-theoretic data governs descent, linking explicit arithmetic of Pell equations to obstructions in $p$-adic cohomology and providing concrete non-descent phenomena for Jacobians over number fields.

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.

Theorems & Definitions (50)

• Theorem 1.1
• Remark 1.3
• Theorem 1.5
• Remark 1.7
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7