A probabilistic model for the distribution of ranks of elliptic curves over $\mathbb{Q}$

TL;DR

This paper introduces a probabilistic model to predict the distribution of elliptic curve ranks over rationals, compares it with existing data and models, and identifies a Selmer bias affecting the data and predictions.

Contribution

It presents a novel probabilistic framework for elliptic curve ranks and highlights the impact of Selmer bias on data and modeling accuracy.

Findings

The model aligns well with empirical data

Selmer bias significantly influences rank distributions

Comparison with previous results validates the model's effectiveness

Abstract

In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and in our models.