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

Published online by Cambridge University Press:  04 December 2007

Steven J. Miller
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, USAsjmiller@math.ohio-state.edu
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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.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2004