monodromy of Gauss-Manin over a Shimura variety This is probably a difficult question. I would like to understand some particular cases and get some references. The rough question is the following: 
Let $X$ be a PEL Shimura variety and $\pi: \mathcal{A} \to X$ the universal abelian scheme over $X$. 
What can be said about the monodromy of the local system $R^1\pi_\ast \mathbb{Q}$? 
For instance, if $Y(N)$ is the modular curve of level $N$ and $\mathcal{E}$ the corresponding universal elliptic curve and we consider the compactification $Y(N) \hookrightarrow X(N)$, what is the local monodromy around a cusp? 
Has this computation been done for Hilbert modular varieties? 
 A: I'm a bit hesitant to answer since I'm not an expert on Shimura varieties, but I suspect that it is a case of "what you see is what you get"; in other words, it shouldn't be too hard to read off the monodromy. Let me do the simplest case
of a fine moduli space of principally polarized abelian varieties of some level. Analytically $X= H_g/\Gamma$,
where $H_g$ is the Siegel upper half plane and $\Gamma \subset Sp_{2g}(\mathbb{Z})$ is a finite index subgroup. The homotopy exact sequence
$$\pi_2(X)=0\to \pi_1(\mathcal{A}_x)=\mathbb{Z}^{2g}\to\pi_1(\mathcal{A})\to \pi_1(X)=\Gamma\to 1$$
splits via the zero section. The induced action of $\Gamma$ on 
$\pi_1(\mathcal{A}_x)=H_1(\mathcal{A}_x)$ is the monodromy of $H_1$, and this is
just the standard symplectic action of $\Gamma$ on $\mathbb{Z}^{2g}$. Dualizing this, will get you the action on $H^1$, i.e. for $R^1\pi_*\mathbb{Z}$. (You can tensor this with $\mathbb{Q}$ if you like.)
I guess (and I am really just guessing) that in  general, $X= K\backslash G/\Gamma$ is a locally symmetric space, and $G$ comes with a representation into some symplectic group. So the mondromy should be the dual of action of $\Gamma$ on some $\mathbb{Z}^{2g}$ via this representation. 
