Moments of the rank of elliptic curves (Q2880083)

Understanding the rank of rational elliptic curves is a major problem of number theory. Many authors have given upper bounds for the average rank, under various hypotheses. The most striking result so far is the recent result of Bhargava and Shankar [\textit{M. Bhargava} and \textit{A. Shankar}, ``Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank \(0\).'' Preprint: \url{arXiv:1007.0052v1} [math.NT]], who showed unconditionally that the average rank in the family of all elliptic curves is at most \(\frac 76\), by bounding the average order of Selmer groups.NEWLINENEWLINEPrevious bounds on the average rank were obtained under GRH and the Birch and Swinnerton-Dyer Conjecture, which predicts the rank of \(E\) to be equal to its \textit{analytic rank}, that is the order of vanishing of \(L(E,s)\) at \(s=1\). Various authors have studied the average analytic rank in families [\textit{D. Goldfeld}, ``Conjectures on elliptic curves over quadratic fields'', Number theory, Proc. Conf., Carbondale 1979, Lect. Notes Math. 751, 108--118 (1979; Zbl 0417.14031)]; [\textit{A. Brumer}, ``The average rank of elliptic curves. I'', Invent. Math. 109, No. 3, 445--472 (1992; Zbl 0783.14019)]; [\textit{D. R. Heath-Brown}, ``The average analytic rank of elliptic curves'', Duke Math. J. 122, No. 3, 591--623 (2004; Zbl 1063.11013)]; [\textit{M. P. Young}, ``Low-lying zeros of families of elliptic curves'', J. Am. Math. Soc. 19, No. 1, 205-250 (2006; Zbl 1086.11032)]. Heath-Brown proved that in the family of all elliptic curves the average rank is bounded by \(2\), and in the family of quadratic twists it is bounded by \(3/2\). The computation of moments of the rank is implicit in his work, and the goal of the paper under review is to make explicit statements about such moments in the family of quadratic twists of a fixed curve.NEWLINENEWLINEMore precisely it is shown that for a fixed elliptic curve \(E:y^2=x^3+ax+b\) and in the range \(k=o_E(\log\log\log x)\), we have NEWLINENEWLINE\[NEWLINE\begin{multlined} NEWLINE\sum_{D} \left[ r(E_D) + \sum_{\gamma_D \neq 0} \left( \frac{\sin(\gamma_D \log x/2)}{\gamma_D \log x/2}\right)^2 \right]^k W\left( \frac D{x^{k/2} (\log x)^{2k+2}}\right) \leq\\ \frac 12 \left[ \left(k+\frac 12+\frac 1{\sqrt 3} \right)^k + \left( k+\frac 12-\frac 1{\sqrt 3}\right)^k +o_{E,W}(1)\right] \sum_D W\left( \frac D{x^{k/2} (\log x)^{2k+2}}\right). \end{multlined}NEWLINE\]NEWLINE Here, \(E_D: Dy^2=x^3+ax+b\), \(r(E_D)\) is the analytic rank of \(E_D\), \(\gamma_D\) runs through the imaginary parts of the zeros of \(L(E_D,s)\) and \(W\) is a test function compactly supported on either \((-1,-1/2)\) or \((1/2,1)\).NEWLINENEWLINEThe authors exploit their main theorem to give various corollaries, such as for example the fact that the proportion of curves of high analytic rank decays at least exponentially fast: NEWLINE\[NEWLINE \sum_{r(E_D)\geq R} W\left(\frac Dx\right) \leq e^{-R/e} \left( O(1)+o_{E,W}((e/R)^{R/e}) \right) \sum_D W\left(\frac Dx\right). NEWLINE\]NEWLINE They also explain its implications on low-lying zeros of \(L(E,s)\), comparing them with the Katz-Sarnak predictions. Finally, the authors combine their result with Silverman's bound on the number of integral points of quasi-minimal models in terms of the rank, yielding interesting conditional bounds on integral points.