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

Published online by Cambridge University Press:  20 November 2018

Matthew P. Young
Matthew P. Young*
Affiliation:
American Institute of Mathematics, Palo Alto, CA 94306-2244, U.S.A. and Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A.
*
myoung@math.tamu.edu
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Abstract

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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 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).

Keywords

11M4111G4011M26
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 5 , 01 October 2010 , pp. 1155 - 1181
Copyright
Copyright © Canadian Mathematical Society 2010

Footnotes

This research was supported by an NSF Mathematical Sciences Post-Doctoral Fellowship and by the American Institute of Mathematics.

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