Investigations of zeros near the central point of elliptic curve L-functions

被引:21
作者
Miller, Steven J.
机构
[1] Brown Univ, Dept Math, Providence, RI 02912 USA
[2] Univ Texas, Dept Appl Math, San Antonio, TX 78249 USA
关键词
elliptic curves; low-lying zeros; n-level statistics; random-matrix theory;
D O I
10.1080/10586458.2006.10128967
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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 Et 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.
引用
收藏
页码:257 / 279
页数:23
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