Combinatorics Seminar

The Katz-Sarnak Density Conjecture and Bounding Central Point Vanishing of L-Functions

Steven Miller, Williams College

Friday, November 5th, 2021
5:00 PM - 6:00 PM
Hylan 1106A (in-person talk)

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 (RMT) seems to model these spacings away from the central point $s=1/2$. While we have an incomplete understanding as to why a correlation exists between RMT and number theory, this interplay has proved useful for conjecturing answers to classical problems. These RMT 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.

Event contact: iosevich at gmail dot com