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

Abstract. We show that the reductions modulo primes of the elliptic curve behave as predicted by the Sato–Tate conjecture, on average over integers a and b such that and where one of the sets 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.