The endomorphism rings of jacobians of cyclic covers of the projective line

Abstract

Suppose that $K$ is a field of characteristic zero, $K_a$ is its algebraic closure, and that $f(x) \in K[x]$ is an irreducible polynomial of degree $n \ge 5$, whose Galois group coincides either with the full symmetric group $\Sn$ or with the alternating group $\An$. Let $p$ be an odd prime, $\Z[\zeta_p]$ the ring of integers in the $p$th cyclotomic field $\Q(\zeta_p)$. Suppose that $C$ is the smooth projective model of the affine curve $y^p\,{=}\,f(x)$ and $J(C)$ is the jacobian of $C$. We prove that the ring $\End(J(C))$ of $K_a$-endomorphisms of $J(C)$ is canonically isomorphic to $\Z[\zeta_p]$