A Random Matrix Model for a Family of Cusp Forms

TL;DR

This paper extends a random matrix model to predict zero statistics of L-functions associated with holomorphic cusp forms, incorporating lower order terms and discretization effects, and verifies its accuracy through experiments.

Contribution

It develops an excised random matrix model for twists of L-functions of holomorphic cusp forms, generalizing previous elliptic curve models and deriving effective matrix sizes.

Findings

The model accurately predicts zero statistics for the family.

No repulsion observed for forms with weight > 2 and principal nebentype.

The model recovers the elliptic curve case as a special instance.

Abstract

The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve $L$-function families. This led to the creation of the excised model of Due\~{n}ez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are well fit by the data. The key ingredients are relating the discretization of central values of the $L$-functions to excising matrices based on the value of the characteristic polynomials at 1 and using lower order terms (in statistics such as the one-level density and pair-correlation) to adjust the matrix size. We extended this model for a family of twists of an $L$-function associated to a given holomorphic cuspidal newform of odd prime level and arbitrary weight. We derive the corresponding "effective" matrix size for a given form by computing the one-level density and pair-correlation statistics for a chosen family of twists, and we show there is no repulsion for forms with weight greater than 2 and principal nebentype. We experimentally verify the accuracy of the model, and as expected, our model recovers the elliptic curve model.