An elliptic curve test of the L-Functions Ratios Conjecture

TL;DR

This paper tests the L-Functions Ratios Conjecture for quadratic twists of elliptic curves, confirming predictions through number theory and developing bounds for quadratic character sums, with implications for modeling zeros near the central point.

Contribution

It provides a rigorous comparison of the Ratios Conjecture's predictions with number theory for elliptic curve families, introducing a generalized bound for quadratic character sums with restricted discriminants.

Findings

Agreement between conjecture predictions and number theory in 1-level density

Development of a generalized Jutila's bound for quadratic characters

Application of results to model zeros near the central point

Abstract

We compare the L-Function Ratios Conjecture's prediction with number theory for the family of quadratic twists of a fixed elliptic curve with prime conductor, and show agreement in the 1-level density up to an error term of size X^{-(1-sigma)/2} for test functions supported in (-sigma, sigma); this gives us a power-savings for \sigma<1. This test of the Ratios Conjecture introduces complications not seen in previous cases (due to the level of the elliptic curve). Further, the results here are one of the key ingredients in the companion paper [DHKMS2], where they are used to determine the effective matrix size for modeling zeros near the central point for this family. The resulting model beautifully describes the behavior of these low lying zeros for finite conductors, explaining the data observed by Miller in [Mil3]. A key ingredient in our analysis is a generalization of Jutila's bound for sums of quadratic characters with the additional restriction that the fundamental discriminant be congruent to a non-zero square modulo a square-free integer M. This bound is needed for two purposes. The first is to analyze the terms in the explicit formula corresponding to characters raised to an odd power. The second is to determine the main term in the 1-level density of quadratic twists of a fixed form on GL_n. Such an analysis was performed by Rubinstein [Rub], who implicitly assumed that Jutila's bound held with the additional restriction on the fundamental discriminants; in this paper we show that assumption is justified.