Cavicchi, Mattia

On the Boundary and Intersection Motives of Genus 2 Hilbert-Siegel Varieties

Doc. Math. 24, 1033-1098 (2019)
DOI: 10.25537/dm.2019v24.1033-1098
Communicated by Thomas Geisser

Summary

We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties \(S_K\) corresponding to the group \(\mathrm{G}\mathrm{Sp}_{4,F}\) over a totally real field \(F\), along with the relative Chow motives \(^\lambda\mathcal{V}\) of abelian type over \(S_K\) obtained from irreducible representations \(V_\lambda\) of \(\mathrm{G}\mathrm{Sp}_{4,F}\). We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight \(\lambda\), potentially generalisable to other families of Shimura varieties, which characterizes the absence of the \textit{middle weights} 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, the absence of these weights allows us to construct Hecke-equivariant Chow motives over \(\mathbb{Q}\), whose realizations equal interior (or intersection) cohomology of \(S_K\) with \(V_{\lambda}\)-coefficients. We give applications to the construction of homological motives associated to automorphic representations.

Mathematics Subject Classification

14G35, 11F46, 11G18, 11F70

Keywords/Phrases

Shimura varieties, Hilbert-Siegel varieties, boundary motive, intersection motive, weight structures, motives for Hilbert-Siegel modular forms

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Affiliation

Cavicchi, Mattia
LAGA, Institut Galilée, Université Paris 13, F-93430 Villetaneuse, France

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