A heuristic for boundedness of ranks of elliptic curves

被引：29

作者：

Park, Jennifer ^{[1
]}

Poonen, Bjorn ^{[2
]}

Voight, John ^{[3
]}

Wood, Melanie Matchett ^{[4
]}

机构：

[1] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

[2] MIT, Dept Math, 77 Massachusetts Ave, Cambridge, MA 02139 USA

[3] Dartmouth Coll, Dept Math, 6188 Kemeny Hall, Hanover, NH 03755 USA

[4] Univ Wisconsin, Dept Math, 480 Lincoln Dr, Madison, WI 53705 USA

来源：

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY | 2019年 / 21卷 / 09期

基金：

加拿大自然科学与工程研究理事会; 美国国家科学基金会;

关键词：

Elliptic curve; rank; Shafarevich-Tate group; CANONICAL HEIGHT; QUADRATIC TWISTS; MODULAR-FORMS; SELMER GROUPS; L-SERIES; POINTS; NUMBER; DENSITY; VALUES; SUBSPACES;

D O I：

10.4171/JEMS/893

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.

引用

页码：2859 / 2903

页数：45

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