A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups

This book studies the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel–Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin–Selberg L-functions for GL(n) × GL(m), where n + m = N. The book carries through the entire program with an eye toward generalizations. The book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.

Download Full-text

Eisenstein cohomology of arithmetic groups. The case GL2

Inventiones mathematicae ◽

10.1007/bf01404673 ◽

1987 ◽

Vol 89 (1) ◽

pp. 37-118 ◽

Cited By ~ 63

Author(s):

G. Harder

Keyword(s):

Arithmetic Groups ◽

Cohomology Of Arithmetic Groups ◽

Eisenstein Cohomology

Download Full-text

Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

Compositio Mathematica ◽

10.1112/s0010437x09004266 ◽

2009 ◽

Vol 146 (1) ◽

pp. 21-57 ◽

Cited By ~ 1

Author(s):

Harald Grobner

Keyword(s):

Algebraic Group ◽

Eisenstein Series ◽

Symplectic Group ◽

Automorphic Forms ◽

General Theorem ◽

Simple Algebraic Group ◽

Ramanujan Conjecture ◽

Eisenstein Cohomology ◽

Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Download Full-text