## 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

## Abstract:

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 $F$.## References

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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: m.f.a.steinmetz@math.leidenuniv.nl
- 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].
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 8739-8767 - MSC (2020): Primary 11F80
- DOI: https://doi.org/10.1090/tran/8794

Dedicated: In memory of Bas Edixhoven