How low can we go? Understanding zeros of $L$-functions near the central point

Spacings between zeros of $L$-functions occur throughout modern number theory, such as in Chebyshev's bias and the class number problem. Montgomery and Dyson discovered in the 1970's that random matrix theory models these spacings. The initial models are insensitive to finitely many zeros, and thus miss the behavior near the central point. This is the most arithmetically interesting place; for example, the Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group equals the order of vanishing of the associated $L$-function there. To investigate the zeros near the central point, Katz and Sarnak developed a new statistic, the $n$-level density; one application is to bound the average order of vanishing at the central point for a given family of $L$-functions by an integral of a weight against some test function $\phi$. After reviewing early results in the subject and describing how these statistics are computed, we discuss as time permits recent progress and ongoing work on several questions. We describe the Excised Orthogonal Ensembles and their success in explaining the observed repulsion of zeros near the central point for families of $L$-functions,and efforts to extend to other families. We discuss an alternative to the Katz-Sarnak expansion for the $n$-level density which facilitate comparisons with random matrix theory, and applications to improving the bounds on high vanishing at the central point.

This work is joint with numerous summer REU students.

Steve Miller