Eisenstein Cohomology for GL<sub>N</sub> and the Special Values of Rankin–Selberg L-Functions
Anantharam Raghuram, Günter Harder
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
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.
Function field, N0, Field of fractions, Natural number, Group scheme, Ring of integers, Basis (linear algebra), Modular form, Linear combination, Triviality (mathematics), Modular symbol, Field extension, Adele ring, Combination, Computation, Group (mathematics), Constant term, Conjecture, Covering space, Free module, Module (mathematics), Embedding, Monograph, L-function, Dimension, Diagram (category theory), Category of groups, Quantity, Eisenstein series, Prime number, Corollary, Addition, Hereditary property, K0, Calculation, Commutative ring, Subgroup, Induced representation, Algebraic number theory, W0, Euler product, Base change, Linear map, Borel subgroup, Spectral sequence, Sheaf (mathematics), Cohomology, Compact group, Algebraic group, Discrete spectrum, Arithmetic group, Permutation, Coefficient, Elaboration, Special case, Theory, Summation, Requirement, Root of unity, Discrete series representation, Ground field, Maximal torus, Irreducible representation, Level structure, Local system, Scalar (physics), Lie algebra, Natural transformation, Galois group, Freydoon Shahidi, Vector space, Dirichlet character, Number theory, Critical value, Triangular matrix, Hecke L-function, Subset, Lie algebra cohomology, Harish-Chandra, Weyl group, Rational number, Standard L-function, Nilradical, Two-dimensional space, SL2(R), Hecke character, Eigenvalues and eigenvectors, Tensor product, Unitary group, Hecke operator, Bearing (navigation), Langlands dual group, Lie group, National Science Foundation, Automorphic form, Reductive group, Theorem, Functor