Rank bias for elliptic curves mod $p$

TL;DR

This paper explores a conjectured bias in the distribution of elliptic curves' ranks and points modulo a prime, drawing parallels with root number biases in modular forms, and suggests the rank bias may be stronger.

Contribution

It introduces a conjecture linking higher rank elliptic curves to more points mod p and compares this bias to root number biases in modular forms, proposing a hierarchy of biases.

Findings

Conjecture that higher rank elliptic curves have more points mod p.

Identification of an analogous bias in modular forms related to root numbers.

Proposal that the rank bias exceeds the root number bias in magnitude.

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.