05094nam a22005895i 4500 978-3-031-41153-3 DE-He213 20251030111457.0 cr nn 008mamaa 240301s2024 sz | s |||| 0|eng d 9783031411533 978-3-031-41153-3 10.1007/978-3-031-41153-3 doi QA241-247.5 PBH bicssc MAT022000 bisacsh PBH thema Getz, Jayce R. author. aut http://id.loc.gov/vocabulary/relators/aut 72297 An Introduction to Automorphic Representations [electronic resource] : With a view toward trace formulae / by Jayce R. Getz, Heekyoung Hahn. 1st ed. 2024. Cham : Springer International Publishing : Imprint: Springer, 2024. XVIII, 609 p. 1 illus. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Graduate Texts in Mathematics, 2197-5612 ; 300 1. 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. The 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. Number theory. 72298 Topological groups. 72299 Lie groups. 72300 Harmonic analysis. 72301 Algebraic geometry. 72302 Number Theory. 72303 Topological Groups and Lie Groups. 72304 Abstract Harmonic Analysis. 72305 Algebraic Geometry. 72306 Hahn, Heekyoung. author. aut http://id.loc.gov/vocabulary/relators/aut 72307 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031411519 Printed edition: 9783031411526 Printed edition: 9783031426100 Graduate Texts in Mathematics, 2197-5612 ; 300 72308 https://doi.org/10.1007/978-3-031-41153-3 ZDB-2-SMA ZDB-2-SXMS 58060 58060