Curves with absolutely irreducible Tate modules over $p$-adic fields

Abstract

Over every number field, curves whose Jacobians have absolutely irreducible $\ell$-adic Tate modules exist in every positive genus and can be chosen hyperelliptic. Over $p$-adic fields, however, inertia and Frobenius can exclude entire genera. We classify the genera that occur for $\ell\ne p$ in terms of the residue characteristic and the cardinality of the residue field. For odd $p$, we also classify the genera occurring for hyperelliptic curves. These genera have relative density zero among those occurring for arbitrary curves, yielding Jacobians whose isogeny classes contain no hyperelliptic Jacobian. For odd $p$ and even residue degree, we obtain further restrictions on the genus when the curve itself is required to have potentially good reduction. Over number fields, we obtain uniform bounds from absolute irreducibility of the Tate module at two finite places of distinct residue characteristics different from $\ell$. The dimension of the abelian variety alone bounds the number of isomorphism classes in its isogeny class and the least degrees of isogenies between them.