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Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. (English) Zbl 1169.11020
The goal of the article under review is to extend the methods of A. Wiles [Ann. Math. (2) 141, No. 3, 443–551 (1995; Zbl 0823.11029)] and R. Taylor, A. Wiles [Ann. Math. (2) 141, No. 3, 553–572 (1995; Zbl 0823.11030)] from \(\mathrm{GL}_2\) to unitary groups of any rank.
The authors work with the disconnected group \(\mathcal{G}_n\), which is the semidirect product of \(\mathrm{GL}_n\times \mathrm{GL}_1\) by the two element group. In this setting the Taylor-Wiles argument carries over well, and the authors prove \(R=T\) type of theorems in the “minimal case” (i.e., they consider deformation problems where the lifts on the inertia groups away from \(\ell\) are completely prescribed, e.g., as unramified as possible away from \(\ell\)). As the authors indicate in their very well-written introduction, there are three key inputs to their proof:
(1) F. Diamond [Invent. Math. 128, No. 2, 379–391 (1997; Zbl 0916.11037)] and K. Fujiwara’s observation that Mazur’s multiplicity one principle is not needed for the Taylor-Wiles argument.
(2) A trick due to C. Skinner and A. Wiles [Duke Math. J. 107, No. 1, 15–25 (2001; Zbl 1016.11017)] which involves a base-change argument to bypass Ribet’s level lowering results.
(3) The proof of the local Langlands conjecture for \(\mathrm{GL}_n\) by the second and third named author, and its compatibility with the global Langlands correspondence.
The authors prove the automorphy of certain Galois representations \(r: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GSp}_n(\mathbb{Z}_\ell)\) under a list of technical conditions. We do not state these hypotheses in detail, instead we give an overview of their content below. We refer the reader to Theorems 4.4.2 – 4.4.3 and Corollary 4.4.4 of the paper under review.
(\(H_I\)) The image of the representation \(\overline{r}:= r \bmod {\ell}\) restricted to \(\text{Gal}(\overline{F}/F(\zeta_\ell))\) should be big in the sense of Definition 2.5.1 of the article, so that the Chebotarev argument in the Taylor-Wiles method works. This condition holds, for example, when the image of \(\overline{r}\) contains \(\mathrm{Sp}_n(\mathbb{F}_\ell)\) (see Lemma 2.5.5).
(\(H_{C+HT}\)) We have \(\ell>n\) and \(r \big{|}_{\text{Gal}(\overline{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)}\) is crystalline with Hodge-Tate weights lying within the interval \([0,n-1]\). This is to ensure that Fontaine-Laffaille theory applies to calculate the local deformation ring at \(\ell\). Furthermore, the Hodge-Tate weights are assumed to be distinct. This, together with the assumption that \(r\) is valued in the symplectic group ensures that \(r\) satisfies the sort of self-duality which is needed for the numerical criterion in Taylor-Wiles method to hold.
(\(H_S\)) For a non-empty auxiliary set of primes \(S\) and for \(\ell \neq q \in S\), the restriction \(r\big{|}_{G_{\mathbb{Q}_q}}\) “looks as if it could correspond to a Steinberg representation under the local Langlands correspondence”. The authors explain that the set \(S\) is required to be non-empty so that relevant automorphic forms may be transferred to and from unitary groups, so that one can attach Galois representations to these automorphic forms.
(\(H_M\)) Outside the auxiliary set \(S\) and away from \(\ell\). the image of \(r\) on the inertia should be finite. This hypothesis has to do with the fact that the authors are working in the minimal case.
In order to be able to remove the minimality condition (\(H_M\)), the authors formulate a conjecture (Conjecture I and the stronger Conjecture II, both in Section 5.3), which is an analogue of Ihara’s Lemma for elliptic modular forms ([Y. Ihara, Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Bombay Colloq. 1973, 161–202 (1975; Zbl 0343.14007)], K. Ribet [Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 503–514 (1984; Zbl 0575.10024)]). If Conjecture I holds, the authors obtain certain level raising results which, much like in [ A. Wiles, loc. cit.], can be used to treat the non-minimal case.
The third named author has developed a new technique in a sequel to this paper [R. Taylor, Publ. Math., Inst. Hautes Étud. Sci. 108, 183–239 (2008; Zbl 1169.11021)], which may be used to treat the non-minimal case without the relying on Conjecture I. The authors indicate that the results of the paper under review are still stronger than that of R. Taylor (loc. cit.) (assuming Conjecture I), as in loc.cit., a Hecke algebra is identified with a universal deformation ring modulo its nilradical.
Overall, this is a very important paper which is written very nicely; it contains very many details on the most recent technology involved in the proofs of modularity lifting theorems.

MSC:
11F80 Galois representations
11G18 Arithmetic aspects of modular and Shimura varieties
11R34 Galois cohomology
Biographic References:
Mann, Russ
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