1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries

Abstract

Following Katz-Sarnak, Iwaniec-Luo-Sarnak, and Rubinstein, we use the 1- and 2-level densities to study the distribution of low lying zeros for one-parameter rational families of elliptic curves over Q(t). Modulo standard conjectures, for small support the densities agree with Katz and Sarnak's predictions. Further, the densities confirm that the curves' L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture. By studying the 2-level densities of some constant sign families, we find the first examples of families of elliptic curves where we can distinguish SO(even) from SO(odd) symmetry.

55 pages, to appear in Compositio Mathematica

References in corpus (1)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries

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• On the behaviour of root numbers in families of elliptic curves
• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions
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• Second moments and the bias conjecture for the family of cubic pencils
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