Endomorphism rings of certain Jacobians in finite characteristic

Yu. G. Zarhin

Sbornik. Mathematics, Tome 193 (2002) no. 8, pp. 1139-1149

Résumé

We prove that, under certain additional assumptions, the endomorphism ring of the Jacobian of a curve $y^\ell=f(x)$ contains a maximal commutative subring isomorphic to the ring of algebraic integers of the $\ell$th cyclotomic field. Here $\ell$ is an odd prime dividing the degree $n$ of the polynomial $f$ and different from the characteristic of the algebraically closed ground field; moreover, $n\geqslant 9$. The additional assumptions stipulate that all coefficients of $f$ lie in some subfield $K$ over which its (the polynomial's) Galois group coincides with either the full symmetric group $S_n$ or with the alternating group $A_n$.