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Modularity lifting theorems for ordinary Galois representations

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Abstract

We generalize results of Clozel, Harris and Taylor by proving modularity lifting theorems for ordinary l-adic Galois representations of any dimension of an imaginary CM or totally real number field. The main theorems are obtained by establishing an \(R^{{{\mathrm{red}}}}={\mathbb {T}}\) theorem over a Hida family. A key part of the proof is to construct appropriate ordinary lifting rings at the primes dividing l and to determine their irreducible components.

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Notes

  1. In fact, the main theorems also prove that r is automorphic of level prime to l if r is crystalline above l. For this statement, we apply the Taylor-Wiles-Kisin method using fixed weight crystalline ordinary lifting rings for v|l. We show that these crystalline rings are irreducible when the residual representation is trivial at each v|l.

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Acknowledgements

I would like to thank my advisor, Richard Taylor, for suggesting this problem and for his invaluable support and assistance. I thank Toby Gee for many useful conversations and for comments on an earlier draft of this paper. I thank Haruzo Hida and Brian Conrad for kindly answering questions, and Florian Herzig for comments on earlier draft of the paper. Finally, I thank the anonymous referee for helpful remarks and suggestions.

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  1. Boston College, Chestnut Hill, USA

    David Geraghty

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  1. David Geraghty
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Correspondence to David Geraghty.

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Communicated by Toby Gee.

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Geraghty, D. Modularity lifting theorems for ordinary Galois representations. Math. Ann. 373, 1341–1427 (2019). https://doi.org/10.1007/s00208-018-1742-4

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  • DOI: https://doi.org/10.1007/s00208-018-1742-4

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