Endomorphisms of Superelliptic Jacobians

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, Z[ζp] the ring of integers in the pth cyclotomic field, Cf,p : y p = f(x) the corresponding superelliptic curve and J(Cf,p) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(Cf,p) coincides with Z[ζp]. The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity. 2000 Math. Subj. Class: Primary 14H40; Secondary 14K05, 11G30, 11G10