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(AM-83), Volume 83 /$fStephen S. Gelbart 210 1$aPrinceton, NJ : $cPrinceton University Press, $d[2016] 210 4$d©1975 215 $a1 online resource (280 pages) 225 0 $aAnnals of Mathematics Studies ;$v262 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a0-691-08156-5 320 $aIncludes bibliographical references and index. 327 $tFrontmatter -- $tPREFACE -- $tCONTENTS -- $t§1. THE CLASSICAL THEORY -- $t§2. AUTOMORPHIC FORMS AND THE DECOMPOSITION OF L2(?SL(2,?)) -- $t§3. AUTOMORPHIC FORMS AS FUNCTIONS ON THE ADELE GROUP OF GL(2) -- $t§4. THE REPRESENTATIONS OF GL(2) OVER LOCAL AND GLOBAL FIELDS -- $t§5 . CUSP FORMS AND REPRESENTATIONS OF THE ADELE GROUP OF GL(2) -- $t§6. HECKE THEORY FOR GL(2) -- $t§7 . THE CONSTRUCTION OF A SPECIAL CLASS OF AUTOMORPHIC FORMS -- $t§ 8 . EISENSTEIN SERIES AND THE CONTINUOUS SPECTRUM -- $t§9. THE TRACE FORMULA FOR GL(2) -- $t§10. AUTOMORPHIC FORMS ON A QUATERNION ALGEBRA -- $tBIBLIOGRAPHY -- $tINDEX 330 $aThis volume investigates the interplay between the classical theory of automorphic forms and the modern theory of representations of adele groups. Interpreting important recent contributions of Jacquet and Langlands, the author presents new and previously inaccessible results, and systematically develops explicit consequences and connections with the classical theory. The underlying theme is the decomposition of the regular representation of the adele group of GL(2). A detailed proof of the celebrated trace formula of Selberg is included, with a discussion of the possible range of applicability of this formula. Throughout the work the author emphasizes new examples and problems that remain open within the general theory.TABLE OF CONTENTS: 1. The Classical Theory 2. Automorphic Forms and the Decomposition of L2(PSL(2,R) 3. Automorphic Forms as Functions on the Adele Group of GL(2) 4. The Representations of GL(2) over Local and Global Fields 5. Cusp Forms and Representations of the Adele Group of GL(2) 6. Hecke Theory for GL(2) 7. The Construction of a Special Class of Automorphic Forms 8. Eisenstein Series and the Continuous Spectrum 9. The Trace Formula for GL(2) 10. Automorphic Forms on a Quaternion Algebr? 410 0$aAnnals of mathematics studies ;$vNumber 83. 606 $aRepresentations of groups 606 $aAutomorphic forms 606 $aLinear algebraic groups 606 $aAdeles 610 $aAbelian extension. 610 $aAbelian group. 610 $aAbsolute value. 610 $aAddition. 610 $aAdditive group. 610 $aAlgebraic group. 610 $aAlgebraic number field. 610 $aAlgebraic number theory. 610 $aAnalytic continuation. 610 $aAnalytic function. 610 $aArbitrarily large. 610 $aAutomorphic form. 610 $aCartan subgroup. 610 $aClass field theory. 610 $aComplex space. 610 $aCongruence subgroup. 610 $aConjugacy class. 610 $aCoprime integers. 610 $aCusp form. 610 $aDifferential equation. 610 $aDimension (vector space). 610 $aDirect integral. 610 $aDirect sum. 610 $aDivision algebra. 610 $aEigenfunction. 610 $aEigenvalues and eigenvectors. 610 $aEisenstein series. 610 $aEuler product. 610 $aExistential quantification. 610 $aExponential function. 610 $aFactorization. 610 $aFinite field. 610 $aFormal power series. 610 $aFourier series. 610 $aFourier transform. 610 $aFuchsian group. 610 $aFunction (mathematics). 610 $aFunction space. 610 $aFunctional equation. 610 $aFundamental unit (number theory). 610 $aGalois extension. 610 $aGlobal field. 610 $aGroup algebra. 610 $aGroup representation. 610 $aHaar measure. 610 $aHarish-Chandra. 610 $aHecke L-function. 610 $aHilbert space. 610 $aHomomorphism. 610 $aInduced representation. 610 $aInfinite product. 610 $aInner automorphism. 610 $aInteger. 610 $aInvariant measure. 610 $aInvariant subspace. 610 $aIrreducible representation. 610 $aL-function. 610 $aLie algebra. 610 $aLinear map. 610 $aMatrix coefficient. 610 $aMellin transform. 610 $aMeromorphic function. 610 $aModular form. 610 $aP-adic number. 610 $aPoisson summation formula. 610 $aPrime ideal. 610 $aPrime number. 610 $aPrincipal series representation. 610 $aProjective representation. 610 $aQuadratic field. 610 $aQuadratic form. 610 $aQuaternion algebra. 610 $aQuaternion. 610 $aReal number. 610 $aRegular representation. 610 $aRepresentation theory. 610 $aRing (mathematics). 610 $aRing of integers. 610 $aScientific notation. 610 $aSelberg trace formula. 610 $aSimple algebra. 610 $aSquare-integrable function. 610 $aSub"ient. 610 $aSubgroup. 610 $aSummation. 610 $aTheorem. 610 $aTheory. 610 $aTheta function. 610 $aTopological group. 610 $aTopology. 610 $aTrace formula. 610 $aTrivial representation. 610 $aUniqueness theorem. 610 $aUnitary operator. 610 $aUnitary representation. 610 $aUniversal enveloping algebra. 610 $aUpper half-plane. 610 $aVariable (mathematics). 610 $aVector space. 610 $aWeil group. 615 0$aRepresentations of groups. 615 0$aAutomorphic forms. 615 0$aLinear algebraic groups. 615 0$aAdeles. 676 $a512/.22 700 $aGelbart$b Stephen S., $045743 801 0$bDE-B1597 801 1$bDE-B1597 906 $aBOOK 912 $a9910154753203321 996 $aAutomorphic Forms on Adele Groups. (AM-83), Volume 83$92788530 997 $aUNINA