{"id":"AutomorphicForm.GLn.IsSmooth","label":"AutomorphicForm.GLn.IsSmooth","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"A function \\(f:\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\times \\operatorname{GL}_n(\\mathbb {R})\\to \\mathbb {C}\\) is smooth if it has the following three properties. \\(f\\) is continuous. For all \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\), the function \\(y\\mapsto f(x,y)\\) is smooth. For all \\(y\\in \\operatorname{GL}_n(\\mathbb {R})\\), the function \\(x\\mapsto f(x,y)\\) is locally constant.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSmooth","latex_link":"sect0004.html#AutomorphicForm.GLn.IsSmooth","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSmooth","lean_decl_name":"AutomorphicForm.GLn.IsSmooth","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"structure IsSmooth (f : GL (Fin n) (FiniteAdeleRing ℤ ℚ) × GL (Fin n) ℝ → ℂ) : Prop where\n  continuous : Continuous f\n  loc_cst (y : GL (Fin n) ℝ) :\n    IsLocallyConstant (fun x ↦ f (x, y))\n  smooth (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) :\n    ContMDiff 𝓘(ℝ, Matrix (Fin n) (Fin n) ℝ) 𝓘(ℝ, ℂ) ∞ (fun y ↦ f (x, y))\n","line":199,"end_line":204,"proof_status":"statement_formalized"}
{"id":"AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","label":"AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["AutomorphicForm.GLn.IsSmooth","instCentreAction","AutomorphicForm.GLn.Weight","AutomorphicForm.GLn.IsSlowlyIncreasing"],"nlp_text":"A smooth function \\(f:\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\times \\operatorname{GL}_n(\\mathbb {R})\\to \\mathbb {C}\\) is an \\(O_n(\\mathbb {R})\\)-automorphic form on \\(\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}})\\) if it satisfies the following five conditions. (periodicity) For all \\(g\\in \\operatorname{GL}_n(\\mathbb {Q})\\), we have \\(f(gx,gy)=f(x,y)\\). (has a finite level) There exists a compact open subgroup \\(U\\subseteq \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\) such that \\(f(xu,y)=f(x,y)\\) for all \\(u\\in U\\), \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\) and \\(y\\in \\operatorname{GL}_n(\\mathbb {R})\\). (weight \\(\\rho \\)) There exists a continuous finite-dimensional irreducible complex representation \\(\\rho \\) of \\(O_n(\\mathbb {R})\\) such that for every \\((x,y)\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}})\\), the set of functions \\(k\\mapsto f(x,yk)\\) span a finite-dimensional complex vector space isomorphic as \\(O_n(\\mathbb {R})\\)-representation to a direct sum of copies of \\(\\rho \\). (has an infinite level) There is an ideal \\(I\\) of the centre \\(Z_n\\) described in the previous section, which has finite complex codimension, and which annihiliates the function \\(y \\mapsto f(x,y)\\) for all \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\). Note that this is a very fancy way of saying “the function satisfies some natural differential equations”. In the case of modular forms, the differential equations are the Cauchy-Riemann equations, which is why modular forms are holomorphic. (growth condition) For every \\(x\\in \\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\), the function \\(y\\mapsto f(x,y)\\) on \\(\\operatorname{GL}_n(\\mathbb {R})\\) is slowly-increasing.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","latex_link":"sect0004.html#AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","lean_decl_name":"AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"/-- Automorphic forms for GL_n/Q with weight ρ. -/\nstructure AutomorphicFormForGLnOverQ (n : ℕ) (ρ : Weight n) where\n  toFun : GL (Fin n) (FiniteAdeleRing ℤ ℚ) × GL (Fin n) ℝ → ℂ\n  is_smooth : IsSmooth toFun\n  is_periodic : ∀ (g : GL (Fin n) ℚ) (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) (y : GL (Fin n) ℝ),\n    toFun (RingHom.GL (algebraMap _ _) _ g * x, RingHom.GL (algebraMap _ _) _ g * y) = toFun (x, y)\n  is_slowly_increasing (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) :\n    IsSlowlyIncreasing (fun y ↦ toFun (x, y))\n  has_finite_level : ∃ U, IsConstantOn U toFun\n  is_finite_cod (x : GL (Fin n) (FiniteAdeleRing ℤ ℚ)) :\n    haveI f : C^∞⟮𝓘(ℝ, _), _; 𝓘(ℝ, ℂ), ℂ⟯ := ⟨fun y ↦ toFun (x, y), is_smooth.smooth x⟩\n    letI m := (actionTensorCAlg'3 (GL (Fin n) ℝ) (Matrix (Fin n) (Fin n) ℝ)).toLinearMap\n    FiniteDimensional ℂ (Z (GL (Fin n) ℝ) (Matrix (Fin n) (Fin n) ℝ) ⧸ (annihilator f).comap m)\n","line":261,"end_line":273,"proof_status":"statement_formalized"}
{"id":"instComplexLieAlgebraAction","label":"instComplexLieAlgebraAction","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["instLieAlgebraAction"],"nlp_text":"This extends to is a natural complex Lie algebra action of the complexification of the real Lie algebra, on the smooth complex functions on \\(\\operatorname{GL}_n(\\mathbb {R})\\).","doc_url":null,"latex_link":"sect0004.html#instComplexLieAlgebraAction","lean_link":null,"lean_decl_name":"instComplexLieAlgebraAction","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"instUniversalEnvelopingAlgebraAction","label":"instUniversalEnvelopingAlgebraAction","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["instComplexLieAlgebraAction"],"nlp_text":"By functoriality, we get an action of the universal enveloping algebra of this complexified Lie algebra on the smooth complex functions.","doc_url":null,"latex_link":"sect0004.html#instUniversalEnvelopingAlgebraAction","lean_link":null,"lean_decl_name":"instUniversalEnvelopingAlgebraAction","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"instCentreAction","label":"instCentreAction","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":["instUniversalEnvelopingAlgebraAction"],"nlp_text":"Thus the centre \\(\\mathbb {Z}_n\\) of this universal enveloping algebra also acts on the smooth complex functions.","doc_url":null,"latex_link":"sect0004.html#instCentreAction","lean_link":null,"lean_decl_name":"instCentreAction","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"instLieAlgebraAction","label":"instLieAlgebraAction","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"There is a natural action of the real Lie algebra of \\(\\operatorname{GL}_n(\\mathbb {R})\\) on the complex vector space of smooth complex-valued functions on \\(\\operatorname{GL}_n(\\mathbb {R})\\).","doc_url":null,"latex_link":"sect0004.html#instLieAlgebraAction","lean_link":null,"lean_decl_name":"instLieAlgebraAction","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"AutomorphicForm.GLn.Weight","label":"AutomorphicForm.GLn.Weight","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"The weight of an automorphic form for \\(\\operatorname{GL}_n/\\mathbb {Q}\\) can be thought of as a finite-dimensional continuous complex representation \\(\\rho \\) of a maximal compact subgroup of \\(\\operatorname{GL}_n(\\mathbb {R})\\), and it’s convenient to choose one (they’re all conjugate) so we choose \\(O_n(\\mathbb {R})\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.Weight","latex_link":"sect0004.html#AutomorphicForm.GLn.Weight","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.Weight","lean_decl_name":"AutomorphicForm.GLn.Weight","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"structure Weight (n : ℕ) where\n  w : preweight n\n  isSimple : Simple w.fdRep\n","line":239,"end_line":241,"proof_status":"statement_formalized"}
{"id":"a0000000139","label":"a0000000139","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"The group \\(\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}}^f)\\) acts (on the left) on the space of automorphic forms for \\(\\operatorname{GL}_n(\\mathbb {A}_{\\mathbb {Q}})\\) by the formula \\((g\\cdot f)(x,y)=f(xg,y)\\).","doc_url":null,"latex_link":"sect0004.html#a0000000139","lean_link":null,"lean_decl_name":"a0000000139","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"AutomorphicForm.GLn.IsSlowlyIncreasing","label":"AutomorphicForm.GLn.IsSlowlyIncreasing","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"We say that a function \\(f:\\operatorname{GL}_n(\\mathbb {R})\\to \\mathbb {C}\\) is slowly-increasing if there’s some real constant \\(C\\) and positive integer \\(n\\) such that \\(|f(M)|\\leq Cs(M)^n\\) for all \\(M\\in \\operatorname{GL}_n(\\mathbb {R})\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSlowlyIncreasing","latex_link":"sect0004.html#AutomorphicForm.GLn.IsSlowlyIncreasing","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/AutomorphicForm.GLn.IsSlowlyIncreasing","lean_decl_name":"AutomorphicForm.GLn.IsSlowlyIncreasing","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch7","source_path":"FLT/GlobalLanglandsConjectures/GLnDefs.lean","lean4_code":"structure IsSlowlyIncreasing (f : GeneralLinearGroup (Fin n) ℝ → ℂ) : Prop where\n  bounded_by : ∃ (C : ℝ) (N : ℕ),\n    ∀ (M : GeneralLinearGroup (Fin n) ℝ),\n    ‖f M‖ ≤ C * (s (M : Matrix (Fin n) (Fin n) ℝ)) ^ N\n","line":211,"end_line":214,"proof_status":"formalized"}
{"id":"a0000000141","label":"a0000000141","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_7.html","dependencies":[],"nlp_text":"This function is well-defined, i.e., it sends a \\(U\\)-invariant form to a \\(U\\)-invariant form which is independent of the choice of \\(g_i\\).","doc_url":null,"latex_link":"sect0004.html#a0000000141","lean_link":null,"lean_decl_name":"a0000000141","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}