Abstract
Let $$(G,{\mathfrak {X}})$$
(
G
,
X
)
be a Shimura datum and K a neat open compact subgroup of $$G(\mathbb {A}_f)$$
G
(
A
f
)
. Under mild hypothesis on $$(G,{\mathfrak {X}})$$
(
G
,
X
)
, the canonical construction associates a variation of Hodge structure on $$\text {Sh}_K(G,{\mathfrak {X}})(\mathbb {C})$$
Sh
K
(
G
,
X
)
(
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 $$\text {Sh}_K(G,{\mathfrak {X}})$$
Sh
K
(
G
,
X
)
and is functorial in $$(G,{\mathfrak {X}})$$
(
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 $$\{(-1,0),(0,-1)\}$$
{
(
-
1
,
0
)
,
(
0
,
-
1
)
}
. If $$(G,{\mathfrak {X}})$$
(
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. Additionally, we provide a classification of Shimura data of PEL-type and demonstrate that the canonical construction is applicable in this context.
Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
