McGill University, December 4 - 7, 2015
We present in this talk a (conjectural) formula for the one-level density of general
one-parameter families of elliptic curves, in term of $n$, the rank of $E$ over $Q(t)$ and the average root number $W_E$ over the family. In the general case, $W_E$ is zero, and the one-level density is given by orthogonal symmetries as predicted by the conjectures of Katz and Sarnak. In the exceptional cases where $W_E \neq 0$, we find that the statistics are given by a weighted sum of even orthogonal and odd orthogonal symmetries. The most dramatic
and counter-intuitive cases occur when $W_E = \pm 1$. In that case, the one-level density exhibits even orthogonal symmetries
when $(-1)^n W_E = 1$ and odd orthogonal symmetries when $(-1)^n W_E = -1$, and there is a shift of the symmetries (between orthogonal odd and orthogonal even) when $n$ is odd.
This is joint work with Robert Lemke Oliver (Stanford University).
This is joint work with Jesse Thorner.