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Explicit Serre weights for two-dimensional Galois representations

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  20 June 2017

Frank Calegari ,
Matthew Emerton ,
Toby Gee  and
Lambros Mavrides
Frank Calegari
Affiliation:
Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA email [email protected]
Matthew Emerton
Affiliation:
Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA email [email protected]
Toby Gee
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email [email protected]
Lambros Mavrides
Affiliation:
Department of Mathematics, King’s College London, London WC2R 2LS, UK email [email protected]
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Abstract

We prove the explicit version of the Buzzard–Diamond–Jarvis conjecture formulated by Dembele et al. (Serre weights and wild ramification in two-dimensional Galois representations, Preprint (2016), arXiv:1603.07708 [math.NT]). More precisely, we prove that it is equivalent to the original Buzzard–Diamond–Jarvis conjecture, which was proved for odd primes (under a mild Taylor–Wiles hypothesis) in earlier work of the third author and coauthors.

Keywords

Galois representationsHilbert modular forms

MSC classification

Primary: 11F80: Galois representations 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 9 , September 2017 , pp. 1893 - 1907
Copyright
© The Authors 2017 

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References

Abrashkin, V. A., The field of norms functor and the Brückner–Vostokov formula , Math. Ann. 308 (1997), 5–19.Google Scholar
Abrashkin, V. and Jenni, R., The field-of-norms functor and the Hilbert symbol for higher local fields , J. Théor. Nombres Bordeaux 24 (2012), 1–39.Google Scholar
Buzzard, K., Diamond, F. and Jarvis, F., On Serre’s conjecture for mod l Galois representations over totally real fields , Duke Math. J. 155 (2010), 105–161.Google Scholar
Chang, S. and Diamond, F., Extensions of rank one (𝜙, 𝛤)-modules and crystalline representations , Compositio Math. 147 (2011), 375–427.Google Scholar
Dembele, L., Diamond, F. and Roberts, D. P., Serre weights and wild ramification in two-dimensional Galois representations, Preprint (2016), arXiv:1603.07708 [math.NT].Google Scholar
Diamond, F. and Savitt, D., Serre weights for locally reducible two-dimensional Galois representations , J. Inst. Math. Jussieu 14 (2015), 639–672.Google Scholar
Gee, T., Liu, T. and Savitt, D., The Buzzard–Diamond–Jarvis conjecture for unitary groups , J. Amer. Math. Soc. 27 (2014), 389–435.Google Scholar
Gee, T., Liu, T. and Savitt, D., The weight part of Serre’s conjecture for GL(2) , Forum Math. Pi e2 (2015), 52pp.Google Scholar
Kisin, M., Moduli of finite flat group schemes, and modularity , Ann. of Math. (2) 170 (2009), 1085–1180.Google Scholar
Laubie, F., Extensions de Lie et groupes d’automorphismes de corps locaux , Compositio Math. 67 (1988), 165–189.Google Scholar
Mavrides, L., On wild ramification in reductions of two-dimensional crystalline Galois representations, PhD thesis, KCL, 2016.Google Scholar
Schmid, H. L., Zyklische algebraische Funktionenkörper vom Grade p n über endlichem Konstantenkörper der Charakteristik p , J. Reine Angew. Math. 175 (1936), 108–123.Google Scholar
Serre, J.-P., Local fields , in Translated from the French by Marvin Jay Greenberg, Graduate Texts in Mathematics, vol. 67 (Springer, New York, 1979), viii+241.Google Scholar
de Shalit, E., Artin–Schreier-Witt extensions as limits of Kummer–Lubin–Tate extensions, and the explicit reciprocity law , in The arithmetic of function fields (Columbus, OH, 1991), Mathematical Sciences Research Institute Publications, vol. 2 (Springer, Berlin, 1992), 413–420.Google Scholar
Wang, X., 2016, in preparation.Google Scholar