On the boundary and intersection motives of genus 2 Hilbert-Siegel varieties

TL;DR

This paper investigates the boundary motives of genus 2 Hilbert-Siegel varieties, establishing criteria for their weight filtration and constructing related Chow motives with applications to automorphic representations.

Contribution

It provides a criterion on the highest weight for the absence of certain weights in boundary motives and constructs Hecke-equivariant Chow motives related to automorphic forms.

Findings

Criterion for absence of weights 0 and 1 in boundary motives

Construction of Hecke-equivariant Chow motives over $\

Applications to motives associated with automorphic representations

Abstract

We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $\mbox{GSp}_{4,F}$ 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 $\mbox{GSp}_{4,F}$. 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 $\lambda$ which characterises the absence of the 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 motives associated to automorphic representations.