Moments of the critical values of families of elliptic curves, with applications

TL;DR

This paper formulates conjectures on the moments of central values of elliptic curve families, revealing their asymptotic behavior and implications for distribution and the Riemann hypothesis, with potential applications in number theory.

Contribution

It introduces new conjectures on the moments of elliptic curve L-values and their derivatives, connecting them to orthogonal families and the Riemann hypothesis.

Findings

Predicted the order of magnitude of moments matches orthogonal L-function families.

Conjectured the distribution of a_p among rank 2 elliptic curves.

Suggested the Riemann hypothesis can follow from first moment estimates.

Abstract

We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family. Furthermore, as arithmetical applications we make a conjecture on the distribution of a_p's amongst all rank 2 elliptic curves, and also show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).