Level One Algebraic Cusp Forms of Classical Groups of Small Rank.

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.

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 semisimple \mathbb Z-forms of the compact groups \mathrm{SO}_7, \mathrm{SO}_8, \mathrm{SO}_9 (and {\mathrm G}_2) and determine Arthur's endoscopic partition of these spaces in all cases. They also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of \mathrm{GL}_n with trivial infinitesimal character, and to vector valued Siegel modular forms of genus 3. A part of the authors' results are conditional to certain expected results in the theory of twisted endoscopy.

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