Beyond the excised ensemble: modelling elliptic curve L-functions with random matrices

TL;DR

This paper refines a random matrix model for elliptic curve L-functions by explaining the unexpectedly low cutoff and incorporating the detailed structure of the L-value distribution, improving the model's accuracy.

Contribution

It introduces a modified excised model that accounts for the distribution peaks of L-values, clarifying the low cutoff phenomenon and enhancing the modeling of elliptic curve L-functions.

Findings

The modified model explains the low cutoff in the excised ensemble.

Including the first peak of the L-value distribution improves model accuracy.

The distribution of L-values is a superposition of multiple peaks.

Abstract

The `excised ensemble', a random matrix model for the zeros of quadratic twist families of elliptic curve $L$-functions, was introduced by Due\~nez, Huynh, Keating, Miller and Snaith. The excised model is motivated by a formula for central values of these $L$-functions in a paper by Kohnen and Zagier. This formula indicates that for a finite set of $L$-functions from a family of quadratic twists, the central values are all either zero or are greater than some positive cutoff. The excised model imposes this same condition on the central values of characteristic polynomials of matrices from $SO(2N)$. Strangely, the cutoff on the characteristic polynomials that results in a convincing model for the $L$-function zeros is significantly smaller than that which we would obtain by naively transferring Kohnen and Zagier's cutoff to the $SO(2N)$ ensemble. In this current paper we investigate a modification to the excised model. It lacks the simplicity of the original excised ensemble, but it serves to explain the reason for the unexpectedly low cutoff in the original excised model. Additionally, the distribution of central $L$-values is `choppier' than the distribution of characteristic polynomials, in the sense that it is a superposition of a series of peaks: the characteristic polynomial distribution is a smooth approximation to this. The excised model didn't attempt to incorporate these successive peaks, only the initial cutoff. Here we experiment with including some of the structure of the $L$-value distribution. The conclusion is that a critical feature of a good model is to associate the correct mass to the first peak of the $L$-value distribution.