Hodge theory and the Mordell-Weil rank of elliptic curves over extensions of function fields

TL;DR

This paper employs Hodge theory to establish a new upper bound on the Mordell-Weil ranks of elliptic curves over function fields after certain Galois extensions, improving previous bounds under specific conditions.

Contribution

It introduces a novel application of Hodge theory to bound Mordell-Weil ranks, extending and refining earlier results by Silverman and Ellenberg in characteristic zero.

Findings

New upper bound on Mordell-Weil ranks

Improved results for disjoint conductor and ramification supports

Applicable to regular geometrically Galois extensions

Abstract

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension are disjoint.