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

Part of: Number theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 August 2016

Jennifer S. Balakrishnan ,
Wei Ho ,
Nathan Kaplan ,
Simon Spicer ,
William Stein  and
James Weigandt
Jennifer S. Balakrishnan
Affiliation:
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom email balakrishnan@maths.ox.ac.uk
Wei Ho
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA email weiho@umich.edu
Nathan Kaplan
Affiliation:
Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA email nckaplan@math.uci.edu
Simon Spicer
Affiliation:
Facebook Inc., 1 Hacker Way, Menlo Park, CA 94025, USA email mlungu@fb.com
William Stein
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA email wstein@uw.edu
James Weigandt
Affiliation:
Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI 02912, USA email james_weigandt@brown.edu

Abstract

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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 $\mathbb{Q}$, ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$, 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.

MSC classification

Primary: 11G05: Elliptic curves over global fields
Secondary: 11-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 351 - 370
Copyright
© The Author(s) 2016 

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