Cambridge Journal of Mathematics

Volume 11 (2023)

Number 1

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

Pages: 1 – 158



Patrick B. Allen (Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

Chandrashekhar Khare (Department of Mathematics, University of California, Los Angeles, Calif., U.S.A.)

Jack A. Thorne (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)


We prove that many representations $\overline{\rho} : \mathrm {Gal}(\overline{K}/K) \to \mathrm {GL}_2 (\mathbb{F}_3)$, 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.


Galois representations, modularity of elliptic curves

2010 Mathematics Subject Classification

Primary 11F80, 11G05. Secondary 11F75.

P.A. was supported by Simons Foundation Collaboration Grant 527275, NSF grant DMS-1902155, and NSERC. Parts of this work were completed while P.A. was a visitor at the Institute for Advanced Study, where he was partially supported by the NSF.

C.K. was partially supported by NSF grant DMS-2200390 and by a Simons Fellowship.

J.T.’s work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714405). This research was begun during the period that J.T. served as a Clay Research Fellow.

Received 3 August 2021

Published 5 June 2023