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

Let A be a g-dimensional abelian variety over Q whose adelic Galois representation has open image in GSp2gZˆ. We investigate the Frobenius fields Q(πp)=End(Ap)⊗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(πp) is a specified number field and, when A is two-dimensional, at which Q(πp) contains a specified real quadratic number field. These investigations continue the investigations of variants of the Lang-Trotter Conjectures on elliptic curves.