Shimura experts, come to rescue!!

Could somebody else, if not ACL, develop his comment?

Thanks Antoine. Could you elaborate a bit more on your comment and give some references? I would really appreciate that!

Thanks Donu. So that it means that you can compute the residues of the Gauss-Manin connection (at least their classes mod $\mathbb{Z}$) on the compactification without knowning anything about the compactification? This seems a bit weird to me