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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Explicit Serre weights for two-dimensional Galois representations over a ramified base
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by Misja F.A. Steinmetz PDF
Trans. Amer. Math. Soc. 375 (2022), 8739-8767 Request permission


Given a totally real number field $F$ and a mod $p$ Galois representation $\rho \colon G_F\to \mathrm {GL}_2(\bar {\mathbf {F}}_p)$, we propose an explicit definition of the set of Serre weights $W(\rho )$ attached to $\rho$. We prove that our explicit definition is equivalent to previous definitions available in the literature. As a consequence we obtain an explicit Serre’s modularity conjecture for Hilbert modular forms over totally real number fields. Our work generalises previous work of Dembélé–Diamond–Roberts and Calegari–Emerton–Gee–Mavrides which together give explicit and equivalent sets of weights when $p$ is unramified in $F$.
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Additional Information
  • Misja F.A. Steinmetz
  • Affiliation: Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA, Leiden, the Netherlands
  • MR Author ID: 1159805
  • ORCID: 0000-0002-7688-7512
  • Email:
  • Received by editor(s): March 29, 2022
  • Published electronically: September 29, 2022
  • Additional Notes: This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1].

  • Dedicated: In memory of Bas Edixhoven
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 8739-8767
  • MSC (2020): Primary 11F80
  • DOI: