  Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelic
Galois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$.
We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} =
\mathbb{Q}( \pi_p )$ 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 this "Frobenius field"
is a specified number field and, when $A$ is two-dimensional, at which the
Frobenius field contains a specified real quadratic number field. These
investigations continue the investigations of variants of the Lang-Trotter
Conjectures on elliptic curves.
