Local-global compatibility for \(l=p\). I. (English. French summary) Zbl 1259.11057
The compatibility between the local and the global Langlands correspondences at places dividing \(\ell\) for the \(\ell\)-adic Galois representations attached to Hilbert modular forms was proved by H. Carayol at places away from \(\ell\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 3, 409–468 (1986; Zbl 0616.10025)] and by T. Saito for places dividing \(\ell\) [Compos. Math. 145, No. 5, 1081–1113 (2009; Zbl 1259.11060)].
The article under review is set in the quite general context of regular algebraic conjugate self-dual cuspidal automorphic representations of \(\mathrm{GL}_n\) over an imaginary CM field. It proves local-global compatibility of the Langlands correspondences at places dividing \(\ell\) assuming a condition coined Shin regular weights, which is stronger than that of regular weights. In a sequel to this article by the same authors [“Local-global compatibility for \(l=p\). II”, arXiv:1105.2242] this assumption is removed. The local-global compatibility at places away from \(\ell\) is the content of A. Caraiani’s article [Duke Math. J. 161, No. 12, 2311–2413 (2012; Zbl 1405.22028)].
Concerning their methods, the authors state that they are essentially an application of the methods of the paper of R. Taylor and T. Yoshida [J. Am. Math. Soc. 20, No. 2, 467–493 (2007; Zbl 1210.11118)] in the context of the paper of S. W. Shin [Ann. Math. (2) 173, No. 3, 1645–1741 (2011; Zbl 1269.11053)].
The article under review is set in the quite general context of regular algebraic conjugate self-dual cuspidal automorphic representations of \(\mathrm{GL}_n\) over an imaginary CM field. It proves local-global compatibility of the Langlands correspondences at places dividing \(\ell\) assuming a condition coined Shin regular weights, which is stronger than that of regular weights. In a sequel to this article by the same authors [“Local-global compatibility for \(l=p\). II”, arXiv:1105.2242] this assumption is removed. The local-global compatibility at places away from \(\ell\) is the content of A. Caraiani’s article [Duke Math. J. 161, No. 12, 2311–2413 (2012; Zbl 1405.22028)].
Concerning their methods, the authors state that they are essentially an application of the methods of the paper of R. Taylor and T. Yoshida [J. Am. Math. Soc. 20, No. 2, 467–493 (2007; Zbl 1210.11118)] in the context of the paper of S. W. Shin [Ann. Math. (2) 173, No. 3, 1645–1741 (2011; Zbl 1269.11053)].
Reviewer: Gabor Wiese (Luxembourg)
MSC:
| 11F80 | Galois representations |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11R39 | Langlands-Weil conjectures, nonabelian class field theory |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
Keywords:
Langlands correspondences; automorphic forms; Galois representations; local-global compatibilityReferences:
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