Effective equidistribution and the Sato-Tate law for families of elliptic curves

TL;DR

This paper establishes effective bounds on the distribution of elliptic curves modulo p, demonstrating how the accuracy of results depends on input error terms and combinatorial analysis, within the context of the Sato-Tate law.

Contribution

It introduces two methods for proving effective equidistribution bounds for elliptic curves, one relying on moments and combinatorics, the other on the Sato-Tate law, improving previous results.

Findings

Effective bounds for elliptic curves modulo p obeying Sato-Tate law

One method improves bounds by saving a logarithm

Results depend on input error terms and combinatorial complexity

Abstract

Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law. We present two methods of proof. Both use the framework of Murty-Sinha; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato-Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be done.