Functoriality of motivic lifts of the canonical construction

Abstract

Let (G,X) be a Shimura datum and K a neat open compact subgroup of $G({\mathbb{A}}_f)$. Under mild hypothesis on (G,X), the canonical construction associates a variation of Hodge structure on ${\text{Sh}}_K(G,X)(\mathbb{C})$ to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over ${\text{Sh}}_K(G,X)$ and is functorial in (G,X). Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representations of Hodge type {(-1,0),(0,-1)}. If (G,X) is equipped with a choice of PEL-datum, Ancona has defined a motivic lift for all representations of G. We show that this is independent of the choice of PEL-datum and give criteria for it to be compatible with base change.

arXiv comment25 pages, v2 has some mostly typographical improvements. To appear in Manuscripta Mathematica

References in corpus (3)

• Decomposition de motifs abeliens
• On higher regulators of Siegel threefolds II: the connection to the special value
• Regulator constants of integral representations of finite groups

Cited by in corpus (3)

• Heegner points in Coleman families
• Algebraic cycles and functorial lifts from $G_2$ to ${\mathrm{PGSp}}_6$
• On the p-adic interpolation of unitary Friedberg--Jacquet periods