Fredholm Theory and Optimal Test Functions for Detecting Central Point Vanishing Over Families of L-functions

TL;DR

This paper develops a mathematical framework using Fredholm theory to optimize test functions for detecting the order of vanishing of L-functions at the central point, improving bounds on average rank across families.

Contribution

It reduces an infinite-dimensional optimization problem to a finite-dimensional one and explicitly solves many cases, providing sharper bounds on average rank.

Findings

Optimized test functions lead to improved bounds on average rank.

Explicit solutions for many optimization problems are provided.

Bounds improve as the support of test functions increases.

Abstract

The Riemann Zeta-Function is the most studied L-function; it's zeroes give information about the prime numbers. We can associate L-functions to a wide array of objects, and in general, the zeroes of these L-functions give information about those objects. For arbitrary L-functions, the order of vanishing at the central point is of particular important. For example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing at the central point of an elliptic curve L-function is the rank of the Mordell-Weil group of that elliptic curve. The Katz-Sarnak Density Conjecture states that this order vanishing and other behavior are well-modeled by random matrices drawn from the classical compact groups. In particular, the conjecture states that an average order vanishing over a family of L-functions can be bounded using only a given weight function and a chosen test function, phi. The conjecture is known for many families when the test functions are suitably restricted. It is natural to ask which test function is best for each family and for each set of natural restrictions on phi. Our main result is a reduction of an otherwise infinite dimensional optimization to a finite-dimensional optimization problem for all families and all sets of restrictions. We explicitly solve many of these optimization problems and compute the improved bound we obtain on average rank. While we do not verify the density conjecture for these new, looser restrictions, with this project, we are able to precisely quantify the benefits of such efforts with respect to average rank. Finally, we are able to show that this bound strictly improves as we increase support.