Restrictions on endomorphism rings of jacobians and their minimal fields of definition

Abstract

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians $J$ for which the Galois group associated to their 2-torsion is insoluble and 'large' (relative to the dimension of $J$). In this paper we examine what happens when this Galois group merely contains an element of 'large' prime order. In doing so we obtain a partial converse to a result by Guralnick and Kedlaya on the endomorphism field.