On Bhargava's heuristics for $\mathbf{GL}_2(\mathbb{F}_p)$-number fields and the number of elliptic curves of bounded conductor

TL;DR

This paper introduces a new model for counting certain number fields related to elliptic curves, which predicts more fields than previous models, especially for conductors with many prime factors, supported by computational evidence.

Contribution

The paper presents a novel model for counting $ extbf{GL}_2( extbf{F}_p)$-number fields with specific local properties, improving predictions over existing heuristics and incorporating effects of Atkin-Lehner involutions.

Findings

The new model predicts more $ extbf{GL}_2( extbf{F}_p)$-number fields than previous models for large prime factor conductors.

Computational evidence supports the accuracy of the new model.

Degeneracies from Atkin-Lehner involutions significantly affect the count of such number fields.

Abstract

We propose a new model for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields having the same local properties as the splitting field of the mod $p$-Galois representation associated with an elliptic curve over the rational numbers. We explain how this new model and Bhargava's local-to-global heuristics for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields both shed light on the problem of estimating the number of elliptic curves over the rational numbers of squarefree conductor $N < X.$ The new model predicts the existence of significantly more $\mathbf{GL}_2(\mathbb{F}_p)$-number fields with the desired local properties than does the local-to-global model when $N$ has a large number of distinct prime factors, owing to degeneracies caused by Atkin-Lehner involutions. We describe computational evidence supporting the new model.