On orthogonal systems of shifts of scaling function on local fields of positive characteristic

One-level density of families of elliptic curves and the Ratios Conjectures

Using the ratios conjectures as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for two families of L-functions attached to elliptic curves. From those closed formulas, we can determine the underlying symmetry types of the families. The one-level scaling density for the first family corresponds to the orthogonal distribution as predicted by the conjectures of Katz and Sarnak, and the one-level scaling density for the second family is the sum of the Dirac distribution and the even orthogonal distribution. This seems to be a new phenomenon, caused …