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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.
Shimura varieties. --- Cohomology operations. --- Number theory. --- Arithmetic groups. --- L-functions. --- Functions, L --- -Number theory --- Group theory --- Number study --- Numbers, Theory of --- Algebra --- Operations (Algebraic topology) --- Algebraic topology --- Varieties, Shimura --- Arithmetical algebraic geometry --- Addition. --- Adele ring. --- Algebraic group. --- Algebraic number theory. --- Arithmetic group. --- Automorphic form. --- Base change. --- Basis (linear algebra). --- Bearing (navigation). --- Borel subgroup. --- Calculation. --- Category of groups. --- Coefficient. --- Cohomology. --- Combination. --- Commutative ring. --- Compact group. --- Computation. --- Conjecture. --- Constant term. --- Corollary. --- Covering space. --- Critical value. --- Diagram (category theory). --- Dimension. --- Dirichlet character. --- Discrete series representation. --- Discrete spectrum. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elaboration. --- Embedding. --- Euler product. --- Field extension. --- Field of fractions. --- Free module. --- Freydoon Shahidi. --- Function field. --- Functor. --- Galois group. --- Ground field. --- Group (mathematics). --- Group scheme. --- Harish-Chandra. --- Hecke L-function. --- Hecke character. --- Hecke operator. --- Hereditary property. --- Induced representation. --- Irreducible representation. --- K0. --- L-function. --- Langlands dual group. --- Level structure. --- Lie algebra cohomology. --- Lie algebra. --- Lie group. --- Linear combination. --- Linear map. --- Local system. --- Maximal torus. --- Modular form. --- Modular symbol. --- Module (mathematics). --- Monograph. --- N0. --- National Science Foundation. --- Natural number. --- Natural transformation. --- Nilradical. --- Permutation. --- Prime number. --- Quantity. --- Rational number. --- Reductive group. --- Requirement. --- Ring of integers. --- Root of unity. --- SL2(R). --- Scalar (physics). --- Sheaf (mathematics). --- Special case. --- Spectral sequence. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Theorem. --- Theory. --- Triangular matrix. --- Triviality (mathematics). --- Two-dimensional space. --- Unitary group. --- Vector space. --- W0. --- Weyl group.
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This volume investigates the interplay between the classical theory of automorphic forms and the modern theory of representations of adele groups. Interpreting important recent contributions of Jacquet and Langlands, the author presents new and previously inaccessible results, and systematically develops explicit consequences and connections with the classical theory. The underlying theme is the decomposition of the regular representation of the adele group of GL(2). A detailed proof of the celebrated trace formula of Selberg is included, with a discussion of the possible range of applicability of this formula. Throughout the work the author emphasizes new examples and problems that remain open within the general theory.TABLE OF CONTENTS: 1. The Classical Theory 2. Automorphic Forms and the Decomposition of L2(PSL(2,R) 3. Automorphic Forms as Functions on the Adele Group of GL(2) 4. The Representations of GL(2) over Local and Global Fields 5. Cusp Forms and Representations of the Adele Group of GL(2) 6. Hecke Theory for GL(2) 7. The Construction of a Special Class of Automorphic Forms 8. Eisenstein Series and the Continuous Spectrum 9. The Trace Formula for GL(2) 10. Automorphic Forms on a Quaternion Algebr?
Number theory --- Representations of groups --- Linear algebraic groups --- Adeles --- Representations of groups. --- Automorphic forms. --- Linear algebraic groups. --- Adeles. --- Nombres, Théorie des --- Formes automorphes --- Automorphic forms --- Algebraic fields --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Automorphic functions --- Forms (Mathematics) --- Group representation (Mathematics) --- Groups, Representation theory of --- Nombres, Théorie des. --- Abelian extension. --- Abelian group. --- Absolute value. --- Addition. --- Additive group. --- Algebraic group. --- Algebraic number field. --- Algebraic number theory. --- Analytic continuation. --- Analytic function. --- Arbitrarily large. --- Automorphic form. --- Cartan subgroup. --- Class field theory. --- Complex space. --- Congruence subgroup. --- Conjugacy class. --- Coprime integers. --- Cusp form. --- Differential equation. --- Dimension (vector space). --- Direct integral. --- Direct sum. --- Division algebra. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Euler product. --- Existential quantification. --- Exponential function. --- Factorization. --- Finite field. --- Formal power series. --- Fourier series. --- Fourier transform. --- Fuchsian group. --- Function (mathematics). --- Function space. --- Functional equation. --- Fundamental unit (number theory). --- Galois extension. --- Global field. --- Group algebra. --- Group representation. --- Haar measure. --- Harish-Chandra. --- Hecke L-function. --- Hilbert space. --- Homomorphism. --- Induced representation. --- Infinite product. --- Inner automorphism. --- Integer. --- Invariant measure. --- Invariant subspace. --- Irreducible representation. --- L-function. --- Lie algebra. --- Linear map. --- Matrix coefficient. --- Mellin transform. --- Meromorphic function. --- Modular form. --- P-adic number. --- Poisson summation formula. --- Prime ideal. --- Prime number. --- Principal series representation. --- Projective representation. --- Quadratic field. --- Quadratic form. --- Quaternion algebra. --- Quaternion. --- Real number. --- Regular representation. --- Representation theory. --- Ring (mathematics). --- Ring of integers. --- Scientific notation. --- Selberg trace formula. --- Simple algebra. --- Square-integrable function. --- Sub"ient. --- Subgroup. --- Summation. --- Theorem. --- Theory. --- Theta function. --- Topological group. --- Topology. --- Trace formula. --- Trivial representation. --- Uniqueness theorem. --- Unitary operator. --- Unitary representation. --- Universal enveloping algebra. --- Upper half-plane. --- Variable (mathematics). --- Vector space. --- Weil group. --- Nombres, Théorie des
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