Abstract
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.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
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5. Deligne, P.: Valeurs de fonctions $$L$$ et périodes d’intégrales, Automorphic forms, representations and $$L$$-functions. In: Proceedings of the Symposium Pure Mathematics, Oregon State University, Corvallis, Ore. (1977). Part 2, Proceedings of the Symposium Pure Mathematics, XXXIII, American Mathematical Society, Providence, RI (1979), With an appendix by N. Koblitz and A. Ogus, pp. 313–346
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