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Galois representations attached to Hilbert-Siegel modular forms. (English) Zbl 1246.11114
The article under review concerns the construction and properties of Galois representations attached to Hilbert-Siegel modular forms of genus \(2\).
More precisely, the main theorem of the article reunites three important results. Let \(F\) be a totally real number field and \(\pi\) a globally generic cuspidal automorphic representation of \(\mathrm{GSp}(4)\) such that \(\pi_\infty\) is regular algebraic and such that \(\pi\) has a Steinberg component at some finite place.
(a) For a fixed embedding \(\iota: \overline{\mathbb{Q}}_\ell \hookrightarrow \mathbb{C}\), the author establishes the existence of a Galois representation \[ \rho_{\pi,\iota}: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GSp}_4(\overline{\mathbb{Q}}_\ell) \] which satisfies local-global compatibility at all places \(v \nmid \ell\). Moreover, local information is provided for the restriction of \(\rho_{\pi,\iota}\) to places above \(\ell\). We note that the local Langlands correspondence for \(\mathrm{GSp}(4)\) was established by W. T. Gan and S. Takeda [“The local Langlands conjecture for \(\mathrm{GSp}(4)\)”, Ann. Math. (2) 173, No. 3, 1841–1882 (2011; Zbl 1230.11063)].
This result generalises and extends work of G. Laumon [“Fonctions zêtas des variétés de Siegel de dimension trois”, Astérisque 302, 1–66 (2005; Zbl 1097.11021)], R. Taylor [“On the \(l\)-adic cohomology of Siegel threefolds”, Invent. Math. 114, No. 2, 289–310 (1993; Zbl 0810.11034)], and R. Weissauer [“Four dimensional Galois representations”, Astérisque 302, 67–150 (2005; Zbl 1097.11027)] for the case of \(F=\mathbb{Q}\).
The main ingredients in the construction are the fundamental work by Harris and Taylor establishing the local Langlands correspondence for \(\mathrm{GL}(n)\) [M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)] and the subsequent work of Taylor and Yoshida on local-global compatibility [R. Taylor and T. Yoshida, “Compatibility of local and global Langlands correspondences”, J. Am. Math. Soc. 20, No. 2, 467–493 (2007; Zbl 1210.11118)]. In his approach the author has to pass to CM-extensions of \(F\); the descend down to \(F\) is realised by a nice patching argument.
(b) Let \(v \nmid \ell\) such that \(\pi_v\) is Iwahori-special and ramified. Then the restriction of \(\rho_{\pi,\iota}\) to inertia at \(v\) is unipotent. Moreover, the article gives precise information on the rank of the corresponding monodromy operator. This establishes parts of conjectures of A. Genestier and J. Tilouine [“Systèmes de Taylor-Wiles pour \(\mathrm{GSp}(4)\)”, Astérisque 302, 177–290 (2005; Zbl 1142.11036)] as well as C. Skinner and E. Urban [“Sur les déformations \(p\)-adiques de certaines représentations automorphes”, J. Inst. Math. Jussieu 5, No. 4, 629–698 (2006; Zbl 1169.11314)].
(c) Let \(v \nmid \ell\) be a finite place where \(\pi_v\) is supercuspidal and not a lift from \(\mathrm{GO}(2,2)\). Then the local Galois representation at \(v\) is irreducible and the author manages to give a precise relation between the Swan conductor and the depth of \(\pi_v\). Moreover, he derives from this a criterion for tameness at \(v\).

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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