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

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).

24 pages

References in corpus (4)

• On the frequency of vanishing of quadratic twists of modular L-functions
• Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves
• Modeling families of L-functions
• On the nonvanishing of elliptic curve L-functions at the central point

Cited by in corpus (1)

• One-level density of families of elliptic curves and the Ratios Conjectures