Effective Sato–Tate distributions for surfaces arising from products of elliptic curves

We prove, with an unconditional effective error bound, the Sato–Tate distributions for two families of surfaces arising from products of elliptic curves, namely a one-parameter family of K3 surfaces and double quadric surfaces. To prove these effective Sato–Tate distributions, we prove an effective form of the joint Sato–Tate distribution for two twist-inequivalent elliptic curves, along with an effective form of the Sato–Tate distribution for an elliptic curve for primes in arithmetic progressions. The former completes the previous work (Thorner in Res Math Sci 8(1):4, 2021) of Thorner by including the cases in which one of the elliptic curves has CM.