This book studies the interplay between the geometry and topology of
locally symmetric spaces, and the arithmetic aspects of the special
values of L-functions.
The authors study 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) x GL(m), where n + m = N. The authors carry
through the entire program with an eye toward generalizations.
This 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.