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

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

2003-10-12

arxiv.org/abs/math/0310159v1

Compositio Mathematica, Vol. 140, Issue 4, July 2004, 952-992

Mathematics

Number Theory

55 pages, to appear in Compositio Mathematica

Scientific paper

10.1112/S0010437X04000582

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.

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