Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions

TL;DR

This paper investigates how zeros at the central point of elliptic curve L-functions influence the distribution of nearby zeros, revealing a repulsion effect that varies with the rank and conductors of the curves.

Contribution

It provides experimental evidence of zero repulsion near the central point in elliptic curve L-functions and analyzes how this behavior depends on rank and conductors, offering new insights into zero distributions.

Findings

Zeros near the central point are repelled more as rank increases.

Repulsion decreases with increasing conductors, suggesting limits to the effect.

Differences between adjacent zeros are statistically independent of rank and family characteristics.

Abstract

We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve E_t in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r+2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q.