Frobenius distribution for quotients of Fermat curves of prime exponent

Abstract

Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of (l,k) being degenerate or not, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the l-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.