Cyclic Covers of Prime Power Degree, Jacobians and Endomorphisms

Suppose K is a field of characteristic zero, Ka is its algebraic closure, f(x) ∈ K[x] is an irreducible polynomial of degree n ≥ 5, whose Galois group coincides either with the full symmetric group Sn or with the alternating group An. Let q be a power prime, Pq(t) = t q −1 t−1 . Let C be the superelliptic curve y = f(x) and J(C) its jacobian. We prove that if p does not divide n then the algebra End(J(C)) ⊗ Q of Kaendomorphisms of J(C) is canonically isomorphic to Q[t]/Pq(t)Q[t].