04879nam a22005415i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050001600172072001600188072002300204072001500227100008300242245014800325250001800473264007500491300004500566336002600611337002600637338003600663347002400699490005300723505089400776520197901670650002603649650003103675650002303706650003003729650003103759650002603790650004603816650003903862650003103901700008503932710003404017773002604051776003604077776003604113776003604149830006004185856004604245912001404291912001504305999001704320978-3-031-41153-3DE-He21320251030111457.0cr nn 008mamaa240301s2024    sz |    s    |||| 0|eng d  a97830314115339978-3-031-41153-37 a10.1007/978-3-031-41153-32doi 4aQA241-247.5 7aPBH2bicssc 7aMAT0220002bisacsh 7aPBH2thema1 aGetz, Jayce R.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut97229713aAn Introduction to Automorphic Representationsh[electronic resource] :bWith a view toward trace formulae /cby Jayce R. Getz, Heekyoung Hahn.  a1st ed. 2024. 1aCham :bSpringer International Publishing :bImprint: Springer,c2024.  aXVIII, 609 p. 1 illus.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aGraduate Texts in Mathematics,x2197-5612 ;v3000 a1. Affine Algebraic Groups -- 2. Adeles -- 3. Discrete Automorphic Representations -- 4. Archimedean Representation Theory -- 5. Representations of Totally Disconnected Groups -- 6. Automorphic Forms -- 7. Unramified Representations -- 8. Nonarchimedean Representation Theory -- 9. The Cuspidal Spectrum -- 10. Einsenstein Series -- 11. Rankin-Selberg L-functions -- 12. Langlands Functoriality -- 13. Known Cases of Global Langlands Functoriality -- 14. Distinction and Period Integrals -- 15. The Cohomology of Locally Symmetric Spaces -- 16. Spectral Sides of the Trace Formulae -- 17. Orbital Integrals -- 18. Simple Trace Formulae -- 19. Applications of Trace Formulae -- A. Groups attached to involutions of algebras -- B. The Iwasawa Decomposition -- C. Poisson Summation -- D. Alternate conventions related to adelic quotients -- Hints to selected exercises -- References -- Index.  aThe goal of this textbook is to introduce and study automorphic representations, objects at the very core of the Langlands Program. It is designed for use as a primary text for either a semester or a year-long course, for the independent study of advanced topics, or as a reference for researchers. The reader is taken from the beginnings of the subject to the forefront of contemporary research. The journey provides an accessible gateway to one of the most fundamental areas of modern mathematics, with deep connections to arithmetic geometry, representation theory, harmonic analysis, and mathematical physics. The first part of the text is dedicated to developing the notion of automorphic representations. Next, it states a rough version of the Langlands functoriality conjecture, motivated by the description of unramified admissible representations of reductive groups over nonarchimedean local fields. The next chapters develop the theory necessary to make the Langlands functoriality conjecture precise. Thus supercuspidal representations are defined locally, cuspidal representations and Eisenstein series are defined globally, and Rankin-Selberg L-functions are defined to give a link between the global and local settings. This preparation complete, the global Langlands functoriality conjectures are stated and known cases are discussed. This is followed by a treatment of distinguished representations in global and local settings. The link between distinguished representations and geometry is explained in a chapter on the cohomology of locally symmetric spaces (in particular, Shimura varieties). The trace formula, an immensely powerful tool in the Langlands Program, is discussed in the final chapters of the book. Simple versions of the general relative trace formulae are treated for the first time in a textbook, and a wealth of related material on algebraic group actions is included. Outlines for several possible courses are provided in the Preface. 0aNumber theory.972298 0aTopological groups.972299 0aLie groups.972300 0aHarmonic analysis.972301 0aAlgebraic geometry.97230214aNumber Theory.97230324aTopological Groups and Lie Groups.97230424aAbstract Harmonic Analysis.97230524aAlgebraic Geometry.9723061 aHahn, Heekyoung.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut9723072 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303141151908iPrinted edition:z978303141152608iPrinted edition:z9783031426100 0aGraduate Texts in Mathematics,x2197-5612 ;v30097230840uhttps://doi.org/10.1007/978-3-031-41153-3  aZDB-2-SMA  aZDB-2-SXMS  c58060d58060