Non-isogenous Superelliptic Jacobians

Let p be an odd prime. Let K be a field of characteristic zero and Ka its algebraic closure. Let n ≥ 5 and m ≥ 5 be integers. Let f(x), h(x) ∈ K[x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group of f is either the full symmetric group Sn or the alternating group An and the Galois group of h is either the full symmetric group Sm or the alternating group Am. Let us consider the superelliptic curves Cf,p : y p = f(x) and Ch,p : y p = h(x). Let J(Cf,p) be the jacobian of Cf,p and J(Ch,p) be the jacobian of Ch,p. Earlier the author proved that J(Cf,p) and J(Ch,p) are absolutely simple abelian varieties. In the present paper we prove that J(Cf,p) and J(Ch,p) are not isogenous over Ka if the splitting fields of f and h are linearly disjoint over K. 2000 Mathematics Subject Classification: Primary 14H40; Secondary 14K05.