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

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

2010-04-16

arxiv.org/abs/1004.2753v2

Journal of Number Theory 131 (2011), no. 1, 25--44

Mathematics

Number Theory

Version 1.1, 24 pages: corrected the interpretation of Birch's moment calculations, added to the literature review of previous

Scientific paper

10.1016/j.jnt.2010.06.013

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.

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