  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.
