We make conjectures on the moments of the central values of the familyof all elliptic curves and on the moments of the first derivative ofthe central values of a large family of positive rank curves. In bothcases the order of magnitude is the same as that of the moments of thecentral values of an orthogonal family of $L$-functions. Notably, wepredict that the critical values of all rank $1$ elliptic curves islogarithmically larger than the rank $1$ curves in the positive rankfamily.Furthermore, as arithmetical applications, we make a conjecture on thedistribution of $a_p$'s amongst all rank $2$ elliptic curves andshow how the Riemann hypothesis can be deduced from sufficientknowledge of the first moment of the positive rank family (based on anidea of Iwaniec)
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