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

Zachary Feng

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

Number Theory 2022-03-03 v1

Authors: Zachary Feng

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 $\varepsilon > 0$ .

Keywords

modular elliptic curves Galois CM fields lower bound proportion imaginary quadratic fields cyclotomic fields

Cite

@article{arxiv.2203.00731,
  title  = {A lower bound on the proportion of modular elliptic curves over Galois CM fields},
  author = {Zachary Feng},
  journal= {arXiv preprint arXiv:2203.00731},
  year   = {2022}
}

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