On Galois representations and Hilbert-Siegel modular forms

Sorensen, Claus M.
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In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local Langlands correspondence for GSp(4) studied recently in a preprint of Gan and Takeda. As a corollary, we relate the rank of the monodromy operator at p to the dimensions of the parahoric fixed spaces at p. The Galois representations are constructed by first passing to GL(4) over a CM extension, then applying the book of Harris-Taylor plus a refinement due to Taylor-Yoshida, and finally descending to F by a delicate patching argument. This is a variation of the techniques used by Blasius and Rogawski in order to attach motives to Hilbert modular forms.
Comment: 43 pages, comments are very welcome
Mathematics - Number Theory, Mathematics - Representation Theory, 11F33, 11F41, 11F70, 11F80