The understanding of various Frobenius actions plays a central role in the study of arithmetic geometry. For an elliptic curve E over the rational numbers, the Sato—Tate conjecture and the Lang—Trotter philosophy provide heuristics for the behavior of the Frobenius endomorphisms of the reductions of E modulo primes. These heuristics predict that certain sets of primes of density zero ought to be infinite for every E. Their higher-dimensional generalizations apply to abelian surfaces—the 2-dimensional analogs of elliptic curves. I will discuss recent results that in certain cases establish the infinitude of such sets via intersection theory on moduli spaces. The talk is based on joint work with Shankar and Maulik—Shankar