# higher dimensional analogues of the Manin-Drinfeld theorem

The Manin-Drinfeld theorem asserts that a divisor on the compact modular curve $X_0(N)$ which is supported on the cusps is torsion.

Equivalently, if $Y_0(N)$ is the open modular curve, the mixed Hodge structure $H^1(Y_0(N), \mathbb{Q})$ splits.

Are there some kind of generalizations of this theorem to higher-dimensional Shimura varieties and their toroidal compactifications?

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Somewhat related: mathoverflow.net/questions/124502/…, where I asked about Manin-Drinfeld for $\operatorname{GL}_2$ over number fields. –  David Loeffler Jan 19 at 18:11

In various texts by Günter Harder he uses the terminology that the Manin-Drinfeld principle holds for the Shimura variety $X$, the coefficient system $\mathbb V$ and the cohomology theory $H(-)$ (usually $\ell$-adic cohomology or mixed Hodge theory) if there is a direct sum decomposition $$H(X,\mathbb V) = H_!(X,\mathbb V) \oplus H_{Eis}(X,\mathbb V)$$ into inner and Eisenstein cohomology. The former is the image of compactly supported cohomology in ordinary cohomology, the latter is the cokernel.