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

Yuri G. Zarhin

Published 2001-03-28, updated 2002-09-08Version 2

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 $A_n$.