View source for Eisenstein cohomology

The basic ingredients for Eisenstein cohomology are a [[Reductive group|reductive group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300301.png" />, an arithmetic subgroup (cf. also [[Arithmetic group|Arithmetic group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300302.png" /> and a rational representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300303.png" /> (cf. also [[Representation of a group|Representation of a group]]). The simplest example is given by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300304.png" />, a congruence subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300305.png" /> with as representation the space of homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300306.png" />,

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300308.png" /> acts by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e1300309.png" />.

Starting from these data one can construct a [[Symmetric space|symmetric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003010.png" />, the locally symmetric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003011.png" /> and a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003013.png" />. In the example one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003015.png" /> is the connected component of the identity of the real points of the centre.

One can consider the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003016.png" />. It is a consequence of reduction theory that they form a finite-dimensional (graded) vector space. It carries some further structure: It can be endowed with an action of the so-called Hecke algebra on it. It is a fundamental problem to understand these cohomology groups as a module under this action of the Hecke algebra.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003017.png" /> is torsion free, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003018.png" /> is a [[Riemannian manifold|Riemannian manifold]] with finite volume and sometimes it even carries the structure of a quasi-projective variety. If, for instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003020.png" /> is a compact Riemann surface with a finite non-empty set of points removed.

Form the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003021.png" />. A method for investigating these cohomology groups is provided by the de Rham isomorphism

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003023.png" /> is the de Rham complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003024.png" />, and this de Rham complex is isomorphic to

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003026.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003027.png" />) is the [[Lie algebra|Lie algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003028.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003029.png" />).

This opens the door for the application of representation-theoretical methods, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003030.png" /> is a module under the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003031.png" />.

However, since the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003032.png" /> is not compact in general, one has to be careful: the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003033.png" /> and the Hilbert space technique of decomposing a module into irreducibles cannot be applied so easily. It is possible to define various subspaces, namely the space of cusp forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003034.png" />, or the space of compactly supported functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003035.png" />. These define subspaces in the [[Cohomology|cohomology]], namely the cuspidal cohomology and the so-called  "inner cohomology"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003036.png" />. Classes in one of theses subspaces vanish  "at infinity" . The cuspidal cohomology can be investigated using classical Hilbert space techniques; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003037.png" /> is a countable sum of irreducible subspaces under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003038.png" />.

One wants to understand the rest of the cohomology. To that end one can embed the locally symmetric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003039.png" /> into the Borel–Serre compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003040.png" />; the [[Sheaf|sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003041.png" /> extends to a local system on this compactification. The Borel–Serre compactification is a manifold with corners; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003042.png" /> be its boundary. Then there is an [[Exact sequence|exact sequence]]

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This raises the question to understand such sequences: What is the cohomology of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003045.png" /> and what is the image of the  "global"  cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003046.png" /> in the cohomology of the boundary?

Understanding the cohomology of the boundary requires understanding the cohomology of groups of lower rank, which is sometimes not so difficult. For instance, in the special example above the boundary consists of a finite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003047.png" /> of circles and

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Eisenstein cohomology is designed to provide some understanding of the restriction mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003049.png" />. One starts from a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003050.png" /> (or a certain space of classes) in the cohomology of the boundary. Viewing it as a class in the [[De Rham cohomology|de Rham cohomology]], one can represent it by a form which is not invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003051.png" />, but which is invariant under the smaller group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003052.png" /> given by the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003053.png" /> with a [[Parabolic subgroup|parabolic subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003054.png" /> which enters in the datum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003055.png" />. Then one forms the sum

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giving a global class. The only difficulty is that this sum need not converge. Hence the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003057.png" /> has to be  "twisted"  by a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003058.png" /> varying in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003059.png" /> and whose real part should be large. Then the sum

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will be convergent and represent a [[Holomorphic function|holomorphic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003061.png" />. Langlands' general theory of Eisenstein series implies that this function has a meromorphic continuation and hence one can  "evaluate"  at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003062.png" />. However, various things may happen. One may encounter a pole or the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003063.png" /> need not be closed. If it is closed, one has to compute its restriction to the boundary.

What happens exactly depends, of course, on the original data. The original form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003064.png" /> should be specified more precisely; for instance, one may assume that it is an eigenform for a certain subalgebra of the Hecke algebra. Then as such it produces certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003065.png" />-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003066.png" /> (cf. also [[L-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003067.png" />-function]]) and the answer to the above questions depends on the behaviour of certain of these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003068.png" />-functions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003069.png" />.

This program has been carried out successfully in some low-dimensional cases, see, for instance, [[#References|[a2]]], [[#References|[a3]]], [[#References|[a7]]]. In the example, the restriction mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003070.png" /> is surjective in degree one and zero in degree zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003071.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003072.png" />, then the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003073.png" /> in degree one has codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003074.png" /> and dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003075.png" /> in degree zero. Actually, in this case the theory of Eisenstein series is not needed, since purely topological arguments are sufficient.

It has been demonstrated in [[#References|[a2]]], [[#References|[a3]]], [[#References|[a7]]] that a detailed understanding of the Eisenstein cohomology may have certain arithmetic implications; for instance, one obtains rationality results for special values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003076.png" />-functions. One may also hope that via the influence of the values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003077.png" />-functions on the structure of the cohomology as a module under the Hecke algebra, some interesting arithmetic objects (mixed motives, unramified field extensions) can be constructed that owe their existence to the (arithmetic) properties of certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003078.png" />-values; see [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]].

Finally, there is the following fundamental and very general theorem of J. Franke [[#References|[a1]]]. Using the Eisenstein series and their residues and derivatives one can define the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003079.png" />. This space can also be characterized as a space of functions that satisfy certain growth conditions and differential equations.

This subspace is  "very small"  and Franke's theorem says that the mapping

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induces an isomorphism in cohomology.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Franke,   "Harmonic analysis in weighted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003082.png" />-spaces"  ''Ann. Sci. École Norm. Sup. (4)'' , '''31'''  (1998)  pp. 181–279</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Harder,   "Eisenstein cohomology of arithmetic groups: The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003083.png" />"  ''Invent. Math.'' , '''89'''  (1987)  pp. 37–118</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Harder,   "Some results on the Eisenstein cohomology of arithmetic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003084.png" />"  J.P. Labesse (ed.)  J. Schwermer (ed.) , ''Cohomology of Arithmetic Groups. Proc. Conf. CIRM'' , ''Lecture Notes in Mathematics'' , '''1447''' , Springer  (1990)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Harder,   "Eisenstein cohomology of arithmetic groups and its applications to number theory" , ''Proc. ICM (Kyoto, 1990)'' , Math. Soc. Japan  (1991)  pp. 779–790</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Harder,   "Eisensteinkohomologie und die Konstruktion gemischter Motive" , ''Lecture Notes in Mathematics'' , '''1562''' , Springer  (1993)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Harder,   R. Pink,   "Modular konstruierte unverzweigte abelsche <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003085.png" />-Erweiterungen von <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130030/e13003086.png" /> und die Struktur ihrer Galoisgruppen"  ''Math. Nachr.'' , '''159'''  (1992)  pp. 83–99</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Schwermer,   "On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties"  ''Forum Math.'' , '''7'''  (1995)  pp. 1–28</TD></TR></table>