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

Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein, James Weigandt

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)351-370
Number of pages20
JournalLMS Journal of Computation and Mathematics
Volume19
Issue numberA
DOIs
StatePublished - 2016
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Computational Theory and Mathematics

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