# title
Moments of the Critical Values of Families of Elliptic Curves, with Applications

# abstract
<jats:title>Abstract</jats:title><jats:p>We make conjectures on the moments of the central values of the family of all elliptic curves and on themoments 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 <jats:italic>L</jats:italic>-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.</jats:p><jats:p>Furthermore, as arithmetical applications, we make a conjecture on the distribution of <jats:italic>a<jats:sub>p</jats:sub></jats:italic>'s amongst all rank 2 elliptic curves and 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).</jats:p>

# publicationOutlet
CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES

# keywords[]
4903 Numerical And Computational Mathematics