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020 _a9783031411533
_9978-3-031-41153-3
024 7 _a10.1007/978-3-031-41153-3
_2doi
050 4 _aQA241-247.5
072 7 _aPBH
_2bicssc
072 7 _aMAT022000
_2bisacsh
072 7 _aPBH
_2thema
100 1 _aGetz, Jayce R.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_972297
245 1 3 _aAn Introduction to Automorphic Representations
_h[electronic resource] :
_bWith a view toward trace formulae /
_cby Jayce R. Getz, Heekyoung Hahn.
250 _a1st ed. 2024.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2024.
300 _aXVIII, 609 p. 1 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aGraduate Texts in Mathematics,
_x2197-5612 ;
_v300
505 0 _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.
520 _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.
650 0 _aNumber theory.
_972298
650 0 _aTopological groups.
_972299
650 0 _aLie groups.
_972300
650 0 _aHarmonic analysis.
_972301
650 0 _aAlgebraic geometry.
_972302
650 1 4 _aNumber Theory.
_972303
650 2 4 _aTopological Groups and Lie Groups.
_972304
650 2 4 _aAbstract Harmonic Analysis.
_972305
650 2 4 _aAlgebraic Geometry.
_972306
700 1 _aHahn, Heekyoung.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_972307
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031411519
776 0 8 _iPrinted edition:
_z9783031411526
776 0 8 _iPrinted edition:
_z9783031426100
830 0 _aGraduate Texts in Mathematics,
_x2197-5612 ;
_v300
_972308
856 4 0 _uhttps://doi.org/10.1007/978-3-031-41153-3
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c58060
_d58060