Moments of the error term in the Sato–Tate law for elliptic curves

Abstract

We derive new bounds for moments of the error in the Sato–Tate law over families of elliptic curves. Our estimates are stronger than those obtained in [4] and [5] for the first and second moment, but this comes at the cost of larger ranges of averaging. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method is different from those used in the above-mentioned papers and builds on recent work by the second-named author and K. Sinha [21] who derived a Central Limit Theorem on the distribution of the errors in the Sato–Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch's identities connect moments of coefficients of Hasse–Weil L-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak's identity is combinatorial in nature.