Galois representations attached to Hilbert-Siegel modular forms

Abstract

This article is a spinoff of the book of Harris and Taylor [HT], in which they prove the local Langlands conjecture for \operatorname{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 \pi on \operatorname{GSp}(4) over a totally real field, such that \pi_{\infty} is regular algebraic (that is, \pi is cohomological). When \pi is globally generic (that is, has a non-vanishing Fourier coefficient), and \pi has a Steinberg component at some finite place, we associate a Galois representation compatible with the local Langlands correspondence for \operatorname{GSp}(4) defined by Gan and Takeda in a recent preprint [GT]. Over \mathbb Q , for \pi 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 \operatorname{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 \pi has nonzero vectors fixed by a non-special maximal compact subgroup at v , the corresponding monodromy operator at v has rank at most one.

Locations

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