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

Abstract

Suppose K is a field of characteristic 0, $K_a$ is its algebraic closure, p is an odd prime. Suppose, $f(x)\in K[x]$ is a polynomial of degree $n\ge 5$ without multiple roots. Let us consider a curve $C:y^p=f(x)$ and its jacobian J(C). It is known that the ring End(J(C)) of all $K_a$-endomorphisms of J(C) contains the ring $Z[{\zeta }_p]$ of integers in the pth cyclotomic field (generated by obvious automorphisms of C). We prove that $End(J(C))=Z[{\zeta }_p]$ if the Galois group of f over K is either the symmetric group $S_n$ or the alternating group $A_n$.

LaTeX2e, 14 pages. The paper will appear in Math. Proc. Cambridge Philos. Soc