Experimental Mathematics

Investigations of Zeros near the Central Point of Elliptic Curve $L$-Functions, with an appendix by Eduardo Dueñez

Steven J. Miller

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. In contrast to 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 $\Q$.

Article information

Source
Experiment. Math., Volume 15, Issue 3 (2006), 257-280.

Dates
First available in Project Euclid: 5 April 2007

Miller, Steven J. Investigations of Zeros near the Central Point of Elliptic Curve $L$-Functions, with an appendix by Eduardo Dueñez. Experiment. Math. 15 (2006), no. 3, 257--280. https://projecteuclid.org/euclid.em/1175789758