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

  Published:2010-06-18
 Printed: Oct 2010
  • Matthew P. Young,
    Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A.
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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 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)
MSC Classifications: 11M41, 11G40, 11M26 show english descriptions Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
$L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
11G40 - $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
 

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