Level one algebraic cusp forms of classical groups of small rank
The authors determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of \mathrm{GL}_n over \mathbb Q of any given infinitesimal character, for essentially all n \leq 8. For this, they compute the dimensions of spaces of level 1 automorphic forms for certain...
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| Main Authors: | , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2015
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| Edition: | 1 |
| Series: | Memoirs of the American Mathematical Society |
| Subjects: | |
| ISBN: | 147041094X, 9781470410940 |
| ISSN: | 0065-9266, 1947-6221 |
| Online Access: | Get full text |
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Table of Contents:
- Introduction -- Polynomial invariants of finite subgroups of compact connected Lie groups -- Automorphic representations of classical groups : review of Arthur’s results -- Determination of <inline-formula content-type="math/mathml"> Π alg ⊥ ( PGL n ) \Pi _\textrm {alg}^\bot (\textrm {PGL}_n) </inline-formula> for <inline-formula content-type="math/mathml"> n ≤ 5 n\leq 5 </inline-formula> -- Description of <inline-formula content-type="math/mathml"> Π disc ( SO 7 ) \Pi _\textrm {disc}(\textrm {SO}_7) </inline-formula> and <inline-formula content-type="math/mathml"> Π<!-- Π --> alg s ( PGL 6 ) \Pi _\textrm {alg}^\textrm {s}(\textrm {PGL}_6) </inline-formula> -- Description of <inline-formula content-type="math/mathml"> Π disc ( SO 9 ) \Pi _\textrm {disc}(\textrm {SO}_9) </inline-formula> and <inline-formula content-type="math/mathml"> Π<!-- Π --> alg s ( PGL 8 ) \Pi _\textrm {alg}^\textrm {s}(\textrm {PGL}_8) </inline-formula> -- Description of <inline-formula content-type="math/mathml"> Π disc ( SO 8 ) \Pi _\textrm {disc}(\textrm {SO}_8) </inline-formula> and <inline-formula content-type="math/mathml"> Π<!-- Π --> alg o ( PGL 8 ) \Pi _\textrm {alg}^\textrm {o}(\textrm {PGL}_8) </inline-formula> -- Description of <inline-formula content-type="math/mathml"> Π disc ( G 2 ) \Pi _\textrm {disc}(\textrm {G}_2) </inline-formula> -- Application to Siegel modular forms -- Adams-Johnson packets -- The Langlands group of <inline-formula content-type="math/mathml"> Z \mathbb {Z} </inline-formula> and Sato-Tate groups -- Tables -- The <inline-formula content-type="math/mathml"> 121 121 </inline-formula> level <inline-formula content-type="math/mathml"> 1 1 </inline-formula> automorphic representations of <inline-formula content-type="math/mathml"> SO 25 \textrm {SO}_{25} </inline-formula> with trivial coefficients
- Cover -- Title page -- Chapter 1. Introduction -- 1.1. A counting problem -- 1.2. Motivations -- 1.3. The main result -- 1.4. Langlands-Sato-Tate groups -- 1.5. The symplectic-orthogonal alternative -- 1.6. Case-by-case description, examples in low motivic weight -- 1.7. Generalizations -- 1.8. Methods and proofs -- 1.9. Application to Borcherds even lattices of rank 25 and determinant 2 -- 1.10. A level 1, non-cuspidal, tempered automorphic representation of \GL₂₈ over \Q with weights 0,1,2,\cdots,27 -- Chapter 2. Polynomial invariants of finite subgroups of compact connected Lie groups -- 2.1. The setting -- 2.2. The degenerate Weyl character formula -- 2.3. A computer program -- 2.4. Some numerical applications -- 2.5. Reliability -- 2.6. A check: the harmonic polynomial invariants of a Weyl group -- Chapter 3. Automorphic representations of classical groups : review of Arthur's results -- 3.1. Classical semisimple groups over \Z -- 3.2. Discrete automorphic representations -- 3.3. The case of Chevalley and definite semisimple \Z-groups -- 3.4. Langlands parameterization of Π_{ }( ) -- 3.5. Arthur's symplectic-orthogonal alternative -- 3.6. The symplectic-orthogonal alternative for polarized algebraic regular cuspidal automorphic representations of \GL_{ } over \Q -- 3.7. Arthur's classification: global parameters -- 3.8. The packet Π( ) of a ∈Ψ_{ }( ) -- 3.9. The character _{ } of _{ } -- 3.10. Arthur's multiplicity formula -- Chapter 4. Determination of Π_{ }^{⊥}(\PGL_{ }) for ≤5 -- 4.1. Determination of Π^{⊥}_{ }(\PGL₂) -- 4.2. Determination of Π_{ }^{ }(\PGL₄) -- 4.3. An elementary lifting result for isogenies -- 4.4. Symmetric square functoriality and Π^{⊥}_{ }(\PGL₃) -- 4.5. Tensor product functoriality and Π_{ }^{ }(\PGL₄) -- 4.6. Λ* functorality and Π_{ }^{ }(\PGL₅)
- Chapter 5. Description of Π_{ }( ₇) and Π_{ }^{ }(\PGL₆) -- 5.1. The semisimple \Z-group ₇ -- 5.2. Parameterization by the infinitesimal character -- 5.3. Endoscopic partition of Π_{ }( ₇) -- 5.4. Conclusions -- Chapter 6. Description of Π_{ }( ₉) and Π_{ }^{ }(\PGL₈) -- 6.1. The semisimple \Z-group ₉ -- 6.2. Endoscopic partition of Π_{ } -- 6.3. Conclusions -- Chapter 7. Description of Π_{ }( ₈) and Π_{ }^{ }(\PGL₈) -- 7.1. The semisimple \Z-group ₈ -- 7.2. Endoscopic partition of Π_{ } -- 7.3. Conclusions -- Chapter 8. Description of Π_{ }( ₂) -- 8.1. The semisimple definite ₂ over \Z -- 8.2. Polynomial invariants for ₂(\Z)⊂ ₂(\R) -- 8.3. Endoscopic classification of Π_{ }( ₂) -- 8.4. Conclusions -- Chapter 9. Application to Siegel modular forms -- 9.1. Vector valued Siegel modular forms of level 1 -- 9.2. Two lemmas on holomorphic discrete series -- 9.3. An example: the case of genus 3 -- Appendix A. Adams-Johnson packets -- A.1. Strong inner forms of compact connected real Lie groups -- A.2. Adams-Johnson parameters -- A.3. Adams-Johnson packets -- A.4. Shelstad's parameterization map -- Appendix B. The Langlands group of \Z and Sato-Tate groups -- B.1. The locally compact group ℒ_{\Z} -- B.2. Sato-Tate groups -- B.3. A list in rank ≤8 -- Appendix C. Tables -- Appendix D. The 121 level 1 automorphic representations of ₂₅ with trivial coefficients -- Bibliography -- Back Cover