Young, Matthew P. Low-lying zeros of families of elliptic curves. (English) Zbl 1086.11032 J. Am. Math. Soc. 19, No. 1, 205-250 (2006). For an elliptic curve \(E\) and an even Schwartz class test function \(\phi\), define\[ D_X(E,\phi)= \sum_{\gamma}\phi\left(\frac{\gamma}{2\pi}\,\log X\right), \] where \(\frac12 + i\gamma\) runs over the non-trivial zeros of \(L(s,E)\). Random matrix theory predicts the average behaviour of \(D_X(E,\phi)\) as \(E\) varies over suitable families of curves, with “size” measured by \(X\).Assuming an appropriate form of the generalized Riemann hypothesis, the average of \(D_X(E,\phi)\) in certain families is computed, when \(\text{supp}(\widehat \phi)\) is suitably bounded. For example, if one takes the family of all curves \(y^2=x^3+ax+b\), with \(|a| \leq X^{1/3}\) and \(|b|\leq X^{1/2}\) then one finds that \(D_X(E,\phi)\) has average \(\widehat\phi(0)+\frac12\,\phi(0)\) if \(\text{supp}(\widehat\phi)\subset (-7/9, 7/9)\). This range is longer than that obtained by A. Brumer [Invent. Math. 109, No. 3, 445–472 (1992; Zbl 0743.14019]] and the reviewer [Duke Math. J. 122, No. 3, 591–623 (2004; Zbl 1063.11013)], who had conditions \(\text{supp}(\widehat\phi)\subset (-5/9,5/9)\) and \(\text{supp}(\widehat\phi)\subset (-2/3,2/3)\), respectively. The improvement comes from more refined exponential sum estimates, and allows one to deduce that the average analytic rank is at most 25/14. Since this is strictly less than 2 one can conclude that, under the Generalized Riemann Hypothesis, the weak form of the Birch-Swinnerton-Dyer Conjecture (that the analytic and algebraic ranks are equal) holds for a positive proportion of all elliptic curves.A number of other families are investigated, namely \(y^2= x^3 + ax + b^2\) (positive rank); \(y^2= x(x- 1)(x + b)\) (torsion \((C_2\times C_2)\); \(y^2+axy -by =x^3\) (torsion \(C_3\)); \(y^2 = x(x^2+ ax+b)\) (torsion \(C_2\)); \(y^2+xy-by=x^3-bx^2\) (torsion \(C_4\)); and \(dy^2=x^3+Ax+B\) with \(d=a^3+Aa+B\) (quadratic twists with positive rank). Reviewer: Roger Heath-Brown (Oxford) Cited in 3 ReviewsCited in 45 Documents MSC: 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11M41 Other Dirichlet series and zeta functions 11G05 Elliptic curves over global fields 11L40 Estimates on character sums Keywords:elliptic curve; \(L\)-function; random matrix theory; zeros; 1-level density; average rank; analytic rank; Birch-Swinnerton-Dyer conjecture Citations:Zbl 0743.14019; Zbl 1063.11013 PDFBibTeX XMLCite \textit{M. P. Young}, J. Am. Math. Soc. 19, No. 1, 205--250 (2006; Zbl 1086.11032) Full Text: DOI arXiv References: [1] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over \?: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843 – 939. · Zbl 0982.11033 [2] Armand Brumer, The average rank of elliptic curves. I, Invent. Math. 109 (1992), no. 3, 445 – 472. · Zbl 0783.14019 [3] Armand Brumer and Joseph H. 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