Heuristics for the arithmetic of elliptic curves

TL;DR

This paper introduces a probabilistic model for the arithmetic of elliptic curves, providing theoretical evidence and predictions about their ranks, including the conjecture that ranks are uniformly bounded over the rationals.

Contribution

It develops a new probabilistic framework for understanding elliptic curve ranks and offers predictions supported by theoretical evidence, advancing the field's understanding of elliptic curve distribution.

Findings

Model suggests most elliptic curves have rank ≤ 21

Provides evidence supporting bounded rank conjecture

Makes predictions about elliptic curve properties

Abstract

This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over $\mathbb{Q}$ have rank $\le 21$, which would imply that the rank is uniformly bounded.