Conditional lower bounds on the distribution of central values in families of \(L\)-functions (Q6595600)

In this paper under review, the authors consider first the family of quadratic twists of an elliptic curve \(E\) defined over \(\mathbb{Q}\) with conductor \(N\). More precisely, let \(d\) denote a fundamental discriminant coprime to \(2N\), and let \(\chi_d=\left( \frac{d}{\cdot} \right)\) denote the associated primitive quadratic character. Let \(E_d\) denote the quadratic twist of \(E\) by \(d\), and its associated \(L\)-function \(L(s, E_d)\). The root number of the completed \(L\)-function is denoted by \(\varepsilon_E(d)\). Let define \N\[\N\mathcal{E}=\left\lbrace d: \text{d is a fundamental discriminant with }(d, 2N) = 1\: \text{and}\: \varepsilon_E(d)=1\right\rbrace\N\]\Nand for any \(X\geq 20\) and \(\alpha<\beta\) \N\[\N\mathcal{N}(X;\alpha, \beta)=\left|\left\lbrace d \in \mathcal{E}:X<|d|\leq 2X \cdot \frac{\log L(1/2, E_d)+\frac{1}{2} \log \log |d|}{\sqrt{\log \log |d|}}\in (\alpha, \beta)\right \rbrace\right|.\N\] \NAssuming the the Generalized Riemann Hypothesis (GRH) for the family of twisted \(L\)-functions \(L(s, E\times \chi)\) for all Dirichlet characters \(\chi\), it is shown that \N\[\N\mathcal{N}(X;\alpha, \beta)\geq \left|\lbrace d \in \mathcal{E}:X<|d|\leq 2X\rbrace\right|\left(\frac{1}{4}\frac{1}{\sqrt{2\pi}} \int_\alpha^\beta e^{-x^2/2} dx +o(1) \right).\N\]\NThe method used to prove the above result seems to be very general and applies to many situations where \(1\)-level densities of low-lying zeros in families have been analyzed and yield a positive proportion of non-vanishing for the central values. Such examples were provided by the pioneering work of \textit{H. Iwaniec} et al. [Publ. Math., Inst. Hautes Étud. Sci. 91, 55--131 (2000; Zbl 1012.11041)]. The authors show that the technique described here refines their non-vanishing corollaries, showing that the non-zero \(L\)-values that are produced have the typical size conjectured by \textit{J. P. Keating} and \textit{N. C. Snaith} in their work [Commun. Math. Phys. 214, No. 1, 91--110 (2000; Zbl 1051.11047)].