higher direct image, Shimura varieties of PEL-type and representations

2011-05-19T14:07:30Z

Let $M=M(G,X) = (M_K)_K$ be a Shimura variety of PEL-type associated to datum $(G,X)$. Let $A$ be the universal abelian scheme over $M(G,X)$ and $a: A \rightarrow M$.

Now, using notations from Milne's article from Ann Arbor ("Canonical models..."), take $\rho$ to be a representation of $G^c$. We can associate to it a vector bundle on $M$.

My question is following: is it possible to express $R^q a_* \mathbb{Q} _l$ in terms of the vector bundles associated to representations of $G^c$ as above?

The question comes from the article of Saito "Hilbert modular forms and p-adic Hodge theory", where he says, that it is easy to see using modular interpretation of $M$ ($M$ is not a general PEL-type Shimura variety in the text) that $R^1 a _* \mathbb{Q} _l$ is isomorphic to a vector bundle constructed as above for the very explicit representation (see the beginning of section 6.1). As I don't see it, I would be glad to see an explanation either for this more specific situation or, if the question in general makes sense, I would also be very happy to see a solution for a general Shimura variety of PEL-type.

higher direct image, Shimura varieties of PEL-type and representations - MathOverflow

higher direct image, Shimura varieties of PEL-type and representations