Sorensen, Claus M.

Galois representations attached to Hilbert-Siegel modular forms

Doc. Math., J. DMV 15, 623-670 (2010)

Summary

Summary: This article is a spinoff of the book of Harris and Taylor [HT], in which they prove the local Langlands conjecture for \GL(n), and its companion paper by Taylor and Yoshida [TY] on local-global compatibility. We record some consequences in the case of genus two Hilbert-Siegel modular forms. In other words, we are concerned with cusp forms π on \GSp(4) over a totally real field, such that π is regular algebraic (that is, π is cohomological). When π is globally generic (that is, has a non-vanishing Fourier coefficient), and π has a Steinberg component at some finite place, we associate a Galois representation compatible with the local Langlands correspondence for \GSp(4) defined by Gan and Takeda in a recent preprint [GT]. Over \Q, for π as above, this leads to a new realization of the Galois representations studied previously by Laumon, Taylor and Weissauer. We are hopeful that our approach should apply more generally, once the functorial lift to \GL(4) is understood, and once the so-called book project is completed. An application of the above compatibility is the following special case of a conjecture stated in [SU]: If π has nonzero vectors fixed by a non-special maximal compact subgroup at v, the corresponding monodromy operator at v has rank at most one.

Mathematics Subject Classification

11F33, 11F41, 11F70, 11F80

Keywords/Phrases

Galois representations, Hilbert-Siegel modular forms, monodromy

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