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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

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up vote 4 down vote accepted

This is perhaps more of a comment than an answer.

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.

However I don't actually know any general result ensuring that the Manin-Drinfeld principle holds - but this may very well be due to my own ignorance. But you could start looking through his book (Eisensteinkohomologie und die Konstruktion gemischter Motive) as well as the various manuscripts on his webpage. You will at least find plenty of stuff of the form "if the Manin-Drinfeld principle holds, then we also know that..."

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Thanks for your answer Dan! I will have a look at Harder's book and papers ;) –  johny23 Jan 19 at 8:42
    
I would also look at the series of papers by M. Harris and S. Zucker - boundary cohomology of Shimura varieties. In the beginning of part III, they stay that it is standard that the cohomology decomposes as written by Dan, but I do not know where this is proved. –  ACL Jan 19 at 9:43
    
The nightmare continues? I love this title! –  johny23 Jan 19 at 10:44

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