The square sieve and a Lang–Trotter question for generic abelian varieties

Abstract Let A be a g -dimensional abelian variety over Q whose adelic Galois representation has open image in GSp 2 g Z ˆ . We investigate the Frobenius fields Q ( π p ) = End ( A p ) ⊗ Q of the reduction of A modulo primes p at which this reduction is ordinary and simple. We obtain conditional and unconditional asymptotic upper bounds on the number of primes at which Q ( π p ) is a specified number field and, when A is two-dimensional, at which Q ( π p ) contains a specified real quadratic number field. These investigations continue the investigations of variants of the Lang–Trotter Conjectures on elliptic curves.