Does the Manin-Drinfeld theorem hold over number fields?

Question

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.)

Answer 1

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

http://mr.crossref.org/iPage/?doi=10.1070%2FSM1980v036n04ABEH001848

See also http://www.mathnet.ru/php/person.phtml?&personid=21921&option_lang=eng .

Answer 2

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.

Answer 3

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.