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Rank bias for elliptic curves mod $p$

Kimball Martin and Thomas Pharis

Vol. 15 (2022), No. 4, 709–726
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