Endomorphism rings of certain Jacobians in finite characteristic

Abstract

We prove that, under certain additional assumptions, the endomorphism ring of the Jacobian of a curve contains a maximal commutative subring isomorphic to the ring of algebraic integers of the th cyclotomic field. Here is an odd prime dividing the degree of the polynomial and different from the characteristic of the algebraically closed ground field; moreover, . The additional assumptions stipulate that all coefficients of lie in some subfield over which its (the polynomial's) Galois group coincides with either the full symmetric group or with the alternating group .