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

TextExtending 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 and Sinha (2009) [MS]; 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.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=faW2iDpe5IE.