Modularity of $\operatorname{GL}_2(\mathbb{F}_p)$-representations over CM fields

Patrick B. Allen; Chandrashekhar Khare; Jack A. Thorne

Modularity of $\operatorname{GL}_2(\mathbb{F}_p)$-representations over CM fields

Number Theory 2022-09-05 v2

Authors: Patrick B. Allen , Chandrashekhar Khare , Jack A. Thorne

Abstract

We prove that many representations $\overline{\rho} : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_2(\mathbb{F}_3)$ , where $K$ is a CM field, arise from modular elliptic curves. We prove similar results when the prime $p = 3$ is replaced by $p = 2$ or $p = 5$ . As a consequence, we prove that a positive proportion of elliptic curves over any CM field not containing a 5th root of unity are modular.

Keywords

modularity Galois representations CM fields elliptic curves GL2 finite fields

Cite

@article{arxiv.1910.12986,
  title  = {Modularity of $\operatorname{GL}_2(\mathbb{F}_p)$-representations over CM fields},
  author = {Patrick B. Allen and Chandrashekhar Khare and Jack A. Thorne},
  journal= {arXiv preprint arXiv:1910.12986},
  year   = {2022}
}

Comments

Minor changes and corrections in response to comments of the referee. This is the accepted manuscript

Modularity of $\operatorname{GL}_2(\mathbb{F}_p)$-representations over CM fields

Abstract

Keywords

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