Abstract
Let F be a totally real field and p an odd prime. We prove an automorphy lifting theorem for geometric representations \(\rho :G_F \rightarrow \mathrm {GL}_2(\overline{\mathbb {Q}}_{p})\) which lift irreducible residual representations \( {\overline{\rho }} \) that arise from Hilbert modular forms. The new result is that we allow the case \(p=5\), \( {\overline{\rho }} \) has projective image \(S_5\cong \mathrm {PGL}_2({\mathbb F}_5)\) and the fixed field of the kernel of the projective representation contains \(\zeta _5\). The usual Taylor–Wiles method does not work in this case as there are elements of dual Selmer that cannot be killed by allowing ramification at Taylor–Wiles primes. These elements arise from our hypothesis and the non-vanishing of \(H^1(\mathrm {PGL}_2({\mathbb F}_5),{{\mathrm{Ad}}}(1))\) where \({{\mathrm{Ad}}}(1)\) is the adjoint of the natural representation of \(\mathrm {GL}_2({\mathbb F}_5)\) twisted by the quadratic character of \(\mathrm {PGL}_2({\mathbb F}_5)\).
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Acknowledgments
We are grateful to Toby Gee and to the anonymous referee for useful comments and corrections. C.K. was supported by NSF Grant DMS-1161671 and by a Humboldt Research Award, and thanks the Tata Institute of Fundamental Research, Mumbai for hospitality during the period in which much of the work on this paper was done. He thanks G. Böckle, F. Diamond, N. Fakhruddin, M. Larsen, D. Prasad, R. Ramakrishna and J.-P. Serre for helpful conversations and correspondence. This research was partially conducted during the period J.T. served as a Clay Research Fellow.
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Khare, C.B., Thorne, J.A. Automorphy of some residually \(S_5\) Galois representations. Math. Z. 286, 399–429 (2017). https://doi.org/10.1007/s00209-016-1766-y
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DOI: https://doi.org/10.1007/s00209-016-1766-y