On the Sato-Tate conjecture on average for some families of elliptic curves

We show that the reductions modulo primes p ≤ x of the elliptic curve E_a,b : Y² = X³ + aX + b behave as predicted by the Sato-Tate conjecture, on average over integers a and b such that a ∈ A and b ∈ B where one of the sets A, B ⊆ ℤ is a centered at the origin interval and the other set is of a rather general structure. These asymptotic formulas generalise previous results of W. D. Banks and the author, which in turn improve several previously known results.