Compositio Mathematica

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

Steven J. Miller a1
a1 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

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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.

(Received October 14 2002)
(Accepted June 2 2003)

Key Words: n-level density; low lying zeros; elliptic curve L-functions; Birch and Swinnerton-Dyer conjecture.

Maths Classification

11M26 (primary); 11G05; 11G40; 11M26 (secondary).