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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Hlavní autori: Chenevier, Gaëtan, Renard, David
Médium: E-kniha Kniha
Jazyk:English
Vydavateľské údaje: Providence, Rhode Island American Mathematical Society 2015
Vydanie:1
Edícia:Memoirs of the American Mathematical Society
Predmet:
Cusp forms (Mathematics)
Forms (Mathematics)
euclidean lattices
classical groups
invariants of finite groups
Langlands group of <inline-formula content-type="math/mathml"> Z {\mathbb {Z}} </inline-formula>
Sato-Tate groups
conductor one
endoscopy
vector-valued Siegel modular forms
Automorphic representations
dimension formulas
compact groups
ISBN:147041094X, 9781470410940
ISSN:0065-9266, 1947-6221
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Abstract 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.
AbstractList 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.
We determine the number of level
Author Chenevier, Gaëtan
Renard, David
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  surname: Chenevier
  fullname: Chenevier, Gaëtan
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  givenname: David
  surname: Renard
  fullname: Renard, David
  email: renard@math.polytechnique.fr
  organization: Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, 91128 Palaiseau Cedex, France
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Keywords euclidean lattices
classical groups
invariants of finite groups
Langlands group of <inline-formula content-type="math/mathml"> Z {\mathbb {Z}} </inline-formula>
Sato-Tate groups
conductor one
endoscopy
vector-valued Siegel modular forms
Automorphic representations
dimension formulas
compact groups
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Notes Includes bibliographical references (p. 117-122)
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Snippet We determine the number of level
The authors determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of \mathrm{GL}_n over \mathbb Q of any given...
The authors determine the number of level $1$, polarized, algebraic regular, cuspidal automorphic representations of $\mathrm{GL}_n$ over $\mathbb Q$ of any...
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proquest
nii
ams
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SubjectTerms Cusp forms (Mathematics)
Forms (Mathematics)
TableOfContents 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
Title Level one algebraic cusp forms of classical groups of small rank
URI https://www.ams.org/memo/1121/
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