One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries

Steven J. Miller

doi:10.1112/S0010437X04000582

Abstract

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Following Katz–Sarnak, Iwaniec–Luo–Sarnak and Rubinstein, we use the one- and two-level densities to study the distribution of low-lying zeros for one-parameter rational families of elliptic curves over $\mathbb{Q}(t)$. 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 two-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.

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One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries

Abstract

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This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries

Abstract

Keywords

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