Geodesic
Moments of the Critical Values of Families of Elliptic Curves, with Applications
Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 1155-1181

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DOI

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).
DOI : 10.4153/CJM-2010-049-5
Mots-clés : 11M41, 11G40, 11M26 
Young, Matthew P. Moments of the Critical Values of Families of Elliptic Curves, with Applications. Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 1155-1181. doi: 10.4153/CJM-2010-049-5
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     title = {Moments of the {Critical} {Values} of {Families} of {Elliptic} {Curves,} with {Applications}},
     journal = {Canadian journal of mathematics},
     pages = {1155--1181},
     year = {2010},
     volume = {62},
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     doi = {10.4153/CJM-2010-049-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-049-5/}
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