Rank bias for elliptic curves mod p

Martin, Kimball; Pharis, Thomas

doi:10.2140/involve.2022.15.709

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

Rank bias for elliptic curves mod $p$

Kimball Martin and Thomas Pharis

Vol. 15 (2022), No. 4, 709–726

DOI: 10.2140/involve.2022.15.709

Abstract

We conjecture that, for a fixed prime $p$ , rational elliptic curves with higher rank tend to have more points mod $p$ . We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.

Keywords

elliptic curves, ranks, counting points mod $p$, modular forms, root numbers, Fourier coefficients

Mathematical Subject Classification

Primary: 11F11, 11F30, 11G05

Milestones

Received: 10 December 2021

Revised: 5 January 2022

Accepted: 6 January 2022

Published: 7 January 2023

Communicated by Ken Ono

Authors

Kimball Martin
Department of Mathematics University of Oklahoma Norman, OK United States

Thomas Pharis
Department of Mathematics Indiana University Bloomington, IN United States

Vol. 15, No. 4, 2022