On Bhargava’s Heuristics for GL2(𝔽 p )-Number Fields and the Number of Elliptic Curves of Bounded Conductor

Abstract

We propose a modular forms-based model for counting -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 modular forms-based model and Bhargava’s local-to-global heuristics for counting -number fields both shed light on the problem of estimating the number of elliptic curves over the rational numbers of squarefree conductor The modular forms-based model predicts the existence of significantly more -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 modular forms-based model.