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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 $$v$$.

##### MSC:
 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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