Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

2016/01/01 by Jennifer S. Balakrishnan, Wei Ho, Nathan O. Kaplan +4 · 1 citation
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Algebraic number #Analytic Number Theory Research #Combinatorics #Computer science #Conductor #Database #Elliptic curve #Geometry #Mathematical analysis #Mathematics #Order (exchange) #Pure mathematics #Rank (graph theory) #Supersingular elliptic curve #math.AG #math.NT #msc:11-04 #msc:11G05

paper · pdf · doi:10.1112/s1461157016000152

published as LMS J. Comput. Math. 19 (2016) 351-370 · 20 pages; to appear in ANTS XII. Data available at http://wstein.org/papers/2016-height/

openalex publication_date 2016/01/01 · arxiv created 2016/05/19 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of n -Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over ℚ , ordered by height. We describe databases of elliptic curves over ℚ , ordered by height, in which we compute ranks and 2 -Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

Citations

• Conjectures on elliptic curves over quadratic fields
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• The average number of elements in the 4-Selmer groups of elliptic curves is 7
• Low-lying zeros of families of elliptic curves
• The average rank of elliptic curves I
• The Average Analytic Rank of Elliptic Curves

Cited by

• Murmurations for elliptic curves ordered by height