A lower bound on the proportion of modular elliptic curves over Galois CM fields

TL;DR

This paper establishes explicit lower bounds on the proportion of elliptic curves that are modular over Galois CM fields, with specific results for imaginary quadratic and cyclotomic fields, advancing understanding of modularity distribution.

Contribution

It provides the first explicit lower bounds on the density of modular elliptic curves over Galois CM fields, including special cases like imaginary quadratic and cyclotomic fields.

Findings

At least 2/5 of elliptic curves are modular over imaginary quadratic fields.

For cyclotomic fields with 5 not dividing n, the proportion approaches 1.

Finite exceptions occur for certain n in cyclotomic fields.

Abstract

We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing $\zeta_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic fields $\mathbb{Q}(\zeta_n)$ with $5\nmid n$, this proportion is at least $1-\varepsilon$ with only finitely many exceptions of $n$, for any choice of $\varepsilon > 0$.