# Does the Manin-Drinfeld theorem hold over number fields?

The Manin-Drinfeld theorem has various equivalent statements. Let $\Gamma$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$. Then:

• for any congruence subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$, the subgroup of $\operatorname{Jac}(X(\Gamma))$ generated by the cusps is finite;
• if $c_1, c_2$ are cusps and $f \in S_2(\Gamma)$ is a Hecke eigenform, the integral $\int_{c_1}^{c_2} f(z) \mathrm{d}z$ is a linear combination of the periods $\Omega_+(f)$ and $\Omega_-(f)$ with coefficients in the field generated by the Hecke eigenvalues of $f$;
• the natural map $H^1_c(Y(\Gamma), \mathbb{Q}) \to H^1(Y(\Gamma), \mathbb{Q})$ has a unique Hecke-equivariant section.

I'm interested in extensions of this to the context of arithmetic quotients of $\mathrm{GL}_2(\mathbb{A}_K)$, where $K$ is a number field. The first statement only makes sense if $K$ is totally real, but the second and third can be formulated for any $K$. Are they true in this generality?

(I have a translation of a 1978 paper by Kurcanov which gives the proof for $K$ an imaginary quadratic field; and I believe there is a more recent paper of Kurcanov that covers CM fields, but I can't read Russian and there doesn't seem to be an English translation.)

• I believe there is a paper by Harder in the Janos Bolyai conference (early 1070?) which deals with these questions. – Venkataramana Mar 14 '13 at 12:04
• Harder's paper contains lots of relevant-looking statements on cohomology, compactly-supported cohomology, and cuspidal cohomology, but he doesn't seem to address this question directly as far as I can see. – David Loeffler Mar 14 '13 at 12:24
• the Hecke action on the image of compactly supported cohomology has eigenvalues of absolute value different from those on the eisenstein cohomology and hence there is a natural hecke equivariant section. I believe this is done in that paper of Harder. – Venkataramana Mar 14 '13 at 12:56
• I have the paper in front of me, and it contains nothing whatsoever about eigenvalues of Hecke operators. – David Loeffler Mar 14 '13 at 13:50

## 3 Answers

Here are links to the translations of several Kurchanov's papers.

http://mr.crossref.org/iPage/?doi=10.1070%2FIM1978v012n03ABEH002002

http://mr.crossref.org/iPage/?doi=10.1070%2FIM1980v014n01ABEH001076

• This is useful, thanks -- I didn't realize translations existed of all these papers. – David Loeffler Mar 14 '13 at 13:51
• You are welcome. Actually, since the end of 1960th all major Russian mathematical journals (Izvestija, MatSbornik, Uspekhi, Functional Analysis, MatZametki, . . .) are translated into English. – Yuri Zarhin Mar 14 '13 at 20:24
• That's good to know. Sadly my university library doesn't subscribe to the translation journals, but I can always get them direct from the Institute of Physics. But does anyone know about the non-CM case? – David Loeffler Mar 14 '13 at 21:07

The `pure thought' version of the Manin-Drinfeld theorem, due to Deligne and Elkik, is to show that the extension of mixed Hodge structures splits.

$$0 \rightarrow H^1(X) \rightarrow H^1(Y) \rightarrow H^2_D(X) \rightarrow H^2(X)$$

Here $X$ is a modular curve, $D$ is set of cusps. (this is explained in a paper of Elkik in Asterisque). Perhaps this argument extends to the more general situation - I know it extends to the Kuga-Satake varieties over modular curves. The arguments of Deligne and Elkik use the fact that the Hecke operators act with different eigenvalues.

Here is the reference to the paper of Elkik:

Elkik, R. Le théorème de Manin-Drinfelʹd. (French) [The Manin-Drinfelʹd theorem] Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183 (1990), 59–67.

Along the line Venkataramana brought up, it's mentioned in section 7.6 of the paper Harmonic analysis in weighted $L_2$ spaces by Franke that if compactly-supported cohomology in the 3rd statement is replaced by the cuspidal part of cohomology (which is by definition a Hecke-summand but only defined over $\mathbb{C}$) and we ask about rationality, then when $G=Res^K_{\mathbb{Q}}GL_n$ for a number field $K$ an affirmative answer was given by Clozel (and Franke proved there the rationality of the $\{P\}$-part of cohomology for any class $\{P\}$ of associate parabolic subgroups). It's also remarked there Harder and Clozel seemed to be the first to mention this rationality generalizes Manin-Drinfel'd.