Galois representations attached to Hilbert-Siegel modular forms
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 \textit{W. T. Gan} and \textit{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 \textit{G. Laumon} [``Fonctions zêtas des variétés de Siegel de dimension trois'', Astérisque 302, 1--66 (2005; Zbl 1097.11021)], \textit{R. Taylor} [``On the \(l\)-adic cohomology of Siegel threefolds'', Invent. Math. 114, No. 2, 289--310 (1993; Zbl 0810.11034)], and \textit{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 \(\mathrm{GL}(n)\) [\textit{M. Harris} and \textit{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 [\textit{R. Taylor} and \textit{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 \textit{A. Genestier} and \textit{J. Tilouine} [``Systèmes de Taylor-Wiles pour \(\mathrm{GSp}(4)\)'', Astérisque 302, 177--290 (2005; Zbl 1142.11036)] as well as \textit{C. Skinner} and \textit{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~\(v\).
scientific article; zbMATH DE number 5827287