2-level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries

Following Katz-Sarnak [KS1], [KS2], Iwaniec-Luo-Sarnak [ILS], and Rubinstein [Ru], we use the 1and 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.