Galois representations for general symplectic groups ArnoKret SugWooShin October 16, 2020 Abstract We prove the existence of GSpin-valued Galois representations correspond- ing to cohomological cuspidal automorphic representations of general sym- plectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard–Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank seven motive whose Galois group is of type G2 in the cohomology of Siegel modular va- rieties of genus three. Under some additional local hypotheses we also show automorphic multiplicity one as well as meromorphic continuation of the spin L-functions. Keywords. Automorphic representations, Galois representations, Langlands correspondence, Shimura varieties Contents 1 Conventions and recollections 14 arXiv:1609.04223v4 [math.NT] 15 Oct 2020 2 Arthur parameters for symplectic groups 18 3 Zariski connectedness of image 23 4 Weak acceptability and connected image 26 A. Kret: Korteweg-de Vries Institute Science Park 105 1090 GE Amsterdam, Netherlands; e- mail:
[email protected] S. W. Shin: Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA // Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea; e-mail:
[email protected] Mathematics Subject Classification (2010): Primary 11R39; Secondary 11F70, 11F80, 11G18 1 2 Arno Kret, Sug Woo Shin 5 GSpin-valued Galois