The derivative of SO(2N + 1) characteristic polynomials and rank 3 elliptic curves

Here we calculate the value distribution of the first derivative of characteristic polynomials of matrices from SO(2N + 1) at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. The connection between the values of random matrix characteristic polynomials and values of the L-functions of families of elliptic curves implies that this calculation in random matrix theory is relevant to the problem of predicting the frequency of rank three curves within these families, since the Birch and Swinnerton-Dyer conjecture relates the value of an L-function and its derivatives to the rank of the associated elliptic curve. This article is based on a talk given at the Isaac Newton Institute for Mathematical Sciences during the “Clay Mathematics Institute Special Week on Ranks of Elliptic Curves and Random Matrix Theory”.