The Lang–Trotter conjecture for products of non-CM elliptic curves

Inspired by the work of Lang–Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $${\mathbb {Q}}$$ and by the subsequent generalization of Cojocaru–Davis–Silverberg–Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over $${\mathbb {Q}}$$ that are not $${\overline{{\mathbb {Q}}}}$$ -isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.