Zero repulsion in families of elliptic curve L-functions and an observation of S. J. Miller

TL;DR

This paper explains why the lowest zero of certain elliptic curve L-functions is repelled from the critical point when the central value is nonzero, revealing phenomena beyond standard heuristics.

Contribution

It provides a theoretical explanation for zero repulsion in elliptic curve L-functions, extending understanding beyond Katz-Sarnak heuristics.

Findings

Lowest zero exhibits repulsion from the critical point when L(1/2, E) ≠ 0

Repulsion also observed in cases of first-order vanishing

Results go beyond standard heuristic explanations

Abstract

We provide a theoretical explanation for an observation of S. J. Miller that if L(s,E) is an elliptic curve L-function for which L(1/2, E) is nonzero, then the lowest lying zero of L(s,E) exhibits a repulsion from the critical point which is not explained by the standard Katz-Sarnak heuristics. We establish a similar result in the case of first-order vanishing.