About different cohomology theories used to study Shimura varieties

Question

The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact support in the ordinary cohomology. In Langlands’ Antwerp paper, he also uses parabolic cohomology (of the “Shimura variety” version of modular curve $G(\mathbb{Q})\backslash \mathcal{H}^\pm\times G(\mathbb{A}_f)/K$) and obtain a Matsushima type decomposition of it. But the Matsushima’s formula (also in Borel-Wallach’s book and J.Arthur’s 1981 Invent paper) uses $L^2$-cohomology of the locally symmetric spaces. In Blasius & Rogawski’s paper in Motives, they use intersection cohomology of (the Baily-Borel-Satake compactification of) certain Shimura varieties and use the Matsushima type decomposition because in this case the $L^2$ cohomology is the same as the intersection cohomology. More recently in Sophie Morel’s work, she also uses intersection cohomology.

I am not familiar with algebraic topology and the various cohomology theories mentioned above. The only intuition I have is, some Shimura varieties are non-compact, and their compactifications are probably singular, so one has to put some effort on the cohomology theory used. So my questions are:

1: What is the original motivation for using parabolic cohomology in the early works of Eichler-Shimura, since this seems strange at first. Is there any conceptual reason that it works?

2: Are the $L^2$ and parabolic cohomologies of the locally symmetric space isomorphic?

3: It seems that intersection cohomology is the most powerful one, so can one override the theory of Eichler-Shimura by using intersection cohomology? If so, is there any reference on this (or it is too trivial to be wrote down in any reference? )

4: In the early works of Eichler-Shimura and Langlands, they used different decompositions of the parabolic cohomology, say of Hodge type and Matsushima type respectively. Are they essentially different approaches (to the construction of $\ell$-adic Galois representations) or they are related in some way?

Answer 1

For Eichler-Shimura, I am not a historian, but I have a guess. I think the first papers of Eichler and Shimura handle only the correspondence between modular forms of weight 2 and the Jacobians of (compactified) modular curves. This is the most natural case to think about because it doesn't require any cohomology at all to define. It's also related to the zeta function, which you can see from the title of Shimura's paper "Correspondances modulaires et les fonctions $\zeta$ de courbes algébriques".

If you want to generalize this to higher weight, you might realize that the higher weight modular forms should be related to $H^1$ of a local system $L$. But $L$ has singularities on the cusps, which means there are multiple ways to extend it to the cusps, or equivalently, multiple cohomology theories for the noncompact modular curve. The first thing to do would be to try the two obvious cohomology theories, ordinary and compactly-supported, and realize that they don't work already in the case of constant coefficients that you understand.

In fact, the desired group ($H^1$ of the compactification) is the quotient of compactly supported $H^1$ by some group, and is a subgroup of the ordinary $H^1$. From this it's not too hard to come up with the definition of parabolic cohomology (or to recognize it if you already knew it).

With regards to Matsushima's formula, his formula relates the cohomology of $G/\Gamma$ to the unitary representations of $G$ appearing in $L^2(G/\Gamma)$, and is generalization of work of Cartan and Hodge in the compact case. When you're looking at unitary representations, an $L^2$ space is a very natural place to look for them, and for obvious reasons an $L^2$ space will be easier to relate to $L^2$-cohomology than to other cohomology theories.

With regards to intersection cohomology, using intersection cohomology is very natural if you know (1) the results in $L^2$-cohomology, (2) the relation between $L^2$-cohomology and intersection cohomology, (3) that intersection cohomology exists in characteristic $p$ and has Galois representation structure while $L^2$-cohomology does not. (2) was conjectured by Zucker in 1982, based on some examples where he could prove it, and the existence of the arithmetic structure on intersection cohomology, and its importance, was proved by BBD in 1983.

Donu has essentially answered your question 3 - yes, intersection cohomology can be used to do Eichler-Shimura theory.

I don't understand your question 4.

Answer 2

Let me just comment about 2. I'll let experts on Shimura varieties answers the rest. For a modular curve $X=\Gamma\backslash \mathcal{H}$, a representation of $\Gamma$ gives rise to a local system $L$ on $X$ (assuming $\Gamma$ is torsion free). As I understand it, parabolic cohomology is the image of $H^1_c(X,L)\to H^1(X,L)$. This coincides with $H^1(\bar X, j_*L)$, where $j:X\to \bar X$ is the nonsingular compactification given by adding cusps. This is isomorphic to intersection cohomology $IH^1(X,L)$ (using the naive, rather than BBD, indexing convention).

Now suppose that $X$ is the quotient of an arbitrary Hermitian symmetric space by an arithmetic group $\Gamma$, and take $L$ as before. $L^2$-cohomology with coefficients in $L$ coincides with intersection cohomology by a theorem of Looijenga-Saper-Stern ("the Zucker conjecture"). I'm not sure how parabolic cohomology would be defined in general, but let's say we use the same definition $im[H_c^i(X,L)\to H^i(X,L)]$. Then this generally won't be isomorphic to intersection cohomology, and hence $L^2$-cohomology. This is because intersection cohomology satisfies Poincaré duality, but parabolic cohomology, in this sense, generally won't.