Pure Math Colloquium

Monday, March 23, 2015 4:00 pm - 4:00 pm EDT (GMT -04:00)

Chantal David, Concordia University

“Zeroes and Zeta Functions and Symmetry: One level density for families of L-functions attached to elliptic curves”

The one-level density is about the behavior of low-lying zeroes of L-functions in families, and it is conjectured by Katz and Sarnak that it is given by the equivalent statistics on groups of random matrices. Contrary to other statistics, as the pair correlation, which are universal (i.e. the same for all L-functions), it is believed that the one-level density will differ depending of the ”symmetry type” of the family (unitary, symplectic, orthogonal, even orthogonal and odd orthogonal). We study the one-level density for various families of L-functions attached to ellip- tic curves, using the ratios conjectures as introduced by Conrey, Farmer and Zirnbauer. From the (conjectural) closed formulas that we obtain, we can determine the underlying symmetry type of the families. This cannot always be done with the classical approach to the one-level density, via the explicit formulas, as results can only be achieved for test functions with Fourier transform of limited support, and the three orthogonal symmetry types are then undistinguishable. But this can be done with the ratio conjectures, with somehow surprising results, shedding new light on ”independent” and ”non-independent” zeroes, and the repulsion phenomenon. 1

MC 2017

**Please Note Room**

Refreshments will be served in MC 5413 at 3:30 p.m. Everyone is welcome to attend.