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

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.