Decomposability and Mordell-Weil ranks of Jacobians using Picard numbers

TL;DR

This paper investigates the decomposability of Jacobians and the variation of Mordell-Weil ranks over number fields by analyzing Picard numbers of K3 surface reductions, providing new bounds and examples.

Contribution

It introduces methods to compute Picard numbers for K3 surface reductions and applies them to study Jacobian decomposability and Mordell-Weil rank variations over different ground fields.

Findings

Examples of surfaces with Picard number jumps at all primes of good reduction.

Bound on the genus of curves with Jacobians isomorphic to products of elliptic curves.

Addresses number field analogues of questions on Jacobian decomposability.

Abstract

We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For example, we give examples of surfaces whose Picard numbers jump in rank at all primes of good reduction using Mordell-Weil groups of Jacobians and show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded. The isomorphism result addresses the number field analogue of some questions of Ekedahl and Serre on decomposability of Jacobians of curves into elliptic curves.