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.
Categories: math.AG math.NT