Cyclic covers of prime power degree, jacobians and endomorphisms

Abstract

Suppose that $K$ is a field of characteristic 0, $K_a$ is its algebraic closure, $p$ is a prime, $q=p^r$ is a power prime. Suppose that $f(x)\in K[x]$ is a polynomial of degree $n>4$ without multiple roots. Let us consider the superelliptic curve $C:y^q=f(x)$ and its jacobian $J(C)$. We study the endomorphism algebra $En{d}^0(J(C))$ of all $K_a$-endomorphisms of $J(C)$. We prove that $En{d}^0(J(C))$ is "as small as possible" if the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$.

33 pages, LaTeX2e