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An Introduction to Automorphic Representations
With a view toward trace formulae
Buch von Heekyoung Hahn (u. a.)
Sprache: Englisch

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Beschreibung
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.
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.
Über den Autor

Jayce R. Getz is Associate Professor of Mathematics at Duke University. His research in number theory spans several subjects --- algebraic geometry, automorphic representations, harmonic analysis, and trace formulae. His monograph with M. Goresky (Institute for Advanced Studies), Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change, was awarded the 2011 Ferran Sunyer i Balaguer Prize.

Heekyoung Hahn is Associate Research Professor of Mathematics at Duke University. Her research covers additive combinatorics, Langlands functoriality, and related issues in the representation theory of algebraic groups.
Zusammenfassung

Provides a detailed yet accessible introduction to the Langlands Program

Covers advanced topics that are rarely treated in textbooks, ex. the trace formula and Galois representations

Serves as a highly readable entry guide to a central, and exciting, area of modern mathematics

Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2024
Fachbereich: Arithmetik & Algebra
Genre: Mathematik, Medizin, Naturwissenschaften, Technik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Reihe: Graduate Texts in Mathematics
Inhalt: xviii
609 S.
1 s/w Illustr.
609 p. 1 illus.
ISBN-13: 9783031411519
ISBN-10: 303141151X
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Hahn, Heekyoung
Getz, Jayce R.
Hersteller: Springer International Publishing
Springer International Publishing AG
Graduate Texts in Mathematics
Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße: 241 x 160 x 37 mm
Von/Mit: Heekyoung Hahn (u. a.)
Erscheinungsdatum: 02.03.2024
Gewicht: 1,212 kg
Artikel-ID: 127243451
Über den Autor

Jayce R. Getz is Associate Professor of Mathematics at Duke University. His research in number theory spans several subjects --- algebraic geometry, automorphic representations, harmonic analysis, and trace formulae. His monograph with M. Goresky (Institute for Advanced Studies), Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change, was awarded the 2011 Ferran Sunyer i Balaguer Prize.

Heekyoung Hahn is Associate Research Professor of Mathematics at Duke University. Her research covers additive combinatorics, Langlands functoriality, and related issues in the representation theory of algebraic groups.
Zusammenfassung

Provides a detailed yet accessible introduction to the Langlands Program

Covers advanced topics that are rarely treated in textbooks, ex. the trace formula and Galois representations

Serves as a highly readable entry guide to a central, and exciting, area of modern mathematics

Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2024
Fachbereich: Arithmetik & Algebra
Genre: Mathematik, Medizin, Naturwissenschaften, Technik
Rubrik: Naturwissenschaften & Technik
Medium: Buch
Reihe: Graduate Texts in Mathematics
Inhalt: xviii
609 S.
1 s/w Illustr.
609 p. 1 illus.
ISBN-13: 9783031411519
ISBN-10: 303141151X
Sprache: Englisch
Ausstattung / Beilage: HC runder Rücken kaschiert
Einband: Gebunden
Autor: Hahn, Heekyoung
Getz, Jayce R.
Hersteller: Springer International Publishing
Springer International Publishing AG
Graduate Texts in Mathematics
Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße: 241 x 160 x 37 mm
Von/Mit: Heekyoung Hahn (u. a.)
Erscheinungsdatum: 02.03.2024
Gewicht: 1,212 kg
Artikel-ID: 127243451
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