Subgroups.lean`

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15 declarations, 0 contain sorry, 0 declared as axioms

The subgroups in the Borel-Jacquet definition of an automorphic form

The subgroups of G(𝔸) = G(𝔸_f) × G(ℝ), for G = GL n / ℚ, that the Borel-Jacquet definition of an automorphic form quantifies over; see FormalConjecturesForMathlib.NumberTheory.AutomorphicForm.BorelJacquet for that definition, and Borel and Jacquet's Corvallis article, the reference below, for its source.

Main declarations

All in the namespace Matrix.GeneralLinearGroup:

diagonalEmbedding and ratDiagonal: the diagonal copy of the rational points Γ = G(ℚ) = GL n ℚ in G(𝔸_f) × G(ℝ), the subgroup of condition (a).
orthogonalSubgroup: the orthogonal group K = O n ℝ inside GL n ℝ, the maximal compact subgroup of condition (b2) — mathlib's Matrix.orthogonalGroup n ℝ, a submonoid of Matrix n n ℝ, transported along the coercion to a subgroup of GL n ℝ.
integralAdeles and integralAdelicSubgroup: the integral adeles Ẑ and the compact open subgroup GL n Ẑ of G(𝔸_f) (isOpen_integralAdelicSubgroup, isCompact_integralAdelicSubgroup), which witnesses condition (b1) for constant automorphic forms.

Implementation notes

Compactness of orthogonalSubgroup n and its maximality (Cartan-Iwasawa-Malcev, which also makes it unique up to conjugacy) are asserted in its docstring but not formalised; of compactness, the boundedness half is proved (abs_coe_le_one_of_mem_orthogonalSubgroup).

References:

A. Borel and H. Jacquet, Automorphic forms and automorphic representations, in Automorphic Forms, Representations and L-functions (Corvallis), Proc. Sympos. Pure Math. 33, Part 1, Amer. Math. Soc. (1979), 189–207; §4.

Imports (8)

Imported by 2 files

def Matrix.GeneralLinearGroup.diagonalEmbedding The diagonal embedding of the rational points G(ℚ) = GL n ℚ into G(𝔸_f) × G(ℝ) = GL n 𝔸ᶠ[ℤ, ℚ] × GL n ℝ, through the inclusions of ℚ into the finite adeles and into ℝ line 75 source
Source documentation
The diagonal embedding of the rational points G(ℚ) = GL n ℚ into G(𝔸_f) × G(ℝ) = GL n 𝔸ᶠ[ℤ, ℚ] × GL n ℝ, through the inclusions of ℚ into the finite adeles and into ℝ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, MonoidHom, Matrix.GeneralLinearGroup, Rat, Prod, IsDedekindDomain.FiniteAdeleRing, Int, Real, Matrix
Proof uses: MonoidHom.prod, Matrix.GeneralLinearGroup.map, Algebra.algebraMap
45 instances, coercions and auxiliary constants left out.
```
noncomputable def diagonalEmbedding : GL n ℚ →* GL n 𝔸ᶠ[ℤ, ℚ] × GL n ℝ :=
…
```
def Matrix.GeneralLinearGroup.ratDiagonal The subgroup Γ = G(ℚ) of G(𝔸_f) × G(ℝ): the range of the diagonal embedding of GL n ℚ line 81 source
Source documentation
The subgroup Γ = G(ℚ) of G(𝔸_f) × G(ℝ): the range of the diagonal embedding of GL n ℚ. Condition (a) for an automorphic form is left invariance under this subgroup.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Subgroup, Prod, Matrix.GeneralLinearGroup, IsDedekindDomain.FiniteAdeleRing, Int, Rat, Real, Matrix
Proof uses: MonoidHom.range, Matrix.GeneralLinearGroup.diagonalEmbedding
34 instances, coercions and auxiliary constants left out.
```
noncomputable def ratDiagonal : Subgroup (GL n 𝔸ᶠ[ℤ, ℚ] × GL n ℝ) :=
…
```
def Matrix.GeneralLinearGroup.orthogonalSubgroup Documented declaration: orthogonalSubgroup line 93 source
Source documentation
The orthogonal group O n ℝ as a subgroup of GL n ℝ: the units of Matrix n n ℝ whose underlying matrix lies in mathlib's Matrix.orthogonalGroup n ℝ, the submonoid of matrices whose transpose is their inverse. This is the maximal compact subgroup of GL n ℝ — up to conjugacy the only one, by the Cartan-Iwasawa-Malcev theorem — and it is the K of the pair (G, K) in the definition of an automorphic form for GL n.
Mathlib's Matrix.orthogonalGroup n ℝ is a Submonoid (Matrix n n ℝ); what the definition of an automorphic form needs is a Subgroup (GL n ℝ), so we transport it along the coercion GL n ℝ → Matrix n n ℝ rather than restate the orthogonality condition.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Subgroup, Matrix.GeneralLinearGroup, Real, Matrix
Proof uses: Subgroup.mk, Submonoid.mk, Subsemigroup.mk, Set.ofPred, Membership.mem, Submonoid, Matrix.orthogonalGroup, Set
25 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
def orthogonalSubgroup (n : Type*) [Fintype n] [DecidableEq n] : Subgroup (GL n ℝ) where
…
```
lemma Matrix.GeneralLinearGroup.mem_orthogonalSubgroup line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Matrix.GeneralLinearGroup, Real, Iff, Membership.mem, Subgroup, Matrix, Matrix.GeneralLinearGroup.orthogonalSubgroup, Submonoid, Matrix.orthogonalGroup
Proof uses: Eq.rec, Eq, Iff.rfl, Set, Set.ofPred, Eq.symm, Set.mem_ofPred_eq
20 instances, coercions and auxiliary constants left out.
lemma Matrix.GeneralLinearGroup.mem_orthogonalSubgroup_iff_mul_transpose line 110 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Matrix.GeneralLinearGroup, Real, Iff, Membership.mem, Subgroup, Matrix, Matrix.GeneralLinearGroup.orthogonalSubgroup, Eq, HMul.hMul, Matrix.transpose, OfNat.ofNat
Proof uses: Iff.trans, Submonoid, Matrix.orthogonalGroup, Matrix.GeneralLinearGroup.mem_orthogonalSubgroup, Matrix.mem_orthogonalGroup_iff
34 instances, coercions and auxiliary constants left out.
lemma Matrix.GeneralLinearGroup.mem_orthogonalSubgroup_iff_transpose_eq_coe_inv line 114 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Matrix.GeneralLinearGroup, Real, Iff, Membership.mem, Subgroup, Matrix, Matrix.GeneralLinearGroup.orthogonalSubgroup, Eq, Matrix.transpose, Inv.inv
Proof uses: Eq.mpr, HMul.hMul, OfNat.ofNat, id, congrArg, propext, Matrix.GeneralLinearGroup.mem_orthogonalSubgroup_iff_mul_transpose, Iff.intro, Eq.symm, Matrix.inv_eq_right_inv, Matrix.GeneralLinearGroup.coe_inv, Eq.refl, Units.mul_inv
27 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Matrix.GeneralLinearGroup.abs_coe_le_one_of_mem_orthogonalSubgroup The entries of an orthogonal matrix are bounded by 1: each row is a unit vector line 124 source
Source documentation
The entries of an orthogonal matrix are bounded by 1: each row is a unit vector. This is the boundedness half of the compactness of orthogonalSubgroup n; see the implementation notes on what is and is not formalised about that.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Matrix.GeneralLinearGroup, Real, Membership.mem, Subgroup, Matrix, Matrix.GeneralLinearGroup.orthogonalSubgroup, LE.le, abs, OfNat.ofNat
Proof uses: Eq.mpr, id, Eq, congrArg, rfl, Finset.sum, Finset.univ, HMul.hMul, Eq.mp, Matrix.transpose, Matrix.one_apply_eq, Iff.mp, Matrix.GeneralLinearGroup.mem_orthogonalSubgroup_iff_mul_transpose, Iff.mpr, abs_le_one_iff_mul_self_le_one, Eq.rec, Finset.single_le_sum, Finset, mul_self_nonneg, Finset.mem_univ
54 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
lemma abs_coe_le_one_of_mem_orthogonalSubgroup {y : GL n ℝ} (hy : y ∈ orthogonalSubgroup n)
    (i j : n) : |(y : Matrix n n ℝ) i j| ≤ 1 := by
…
```
Mentions: Matrix.GeneralLinearGroup.orthogonalSubgroup
instance Matrix.GeneralLinearGroup.(instance at line 141) line 141 source
Source documentation
The v-adic integers of ℚ are compact: they are homeomorphic to ℤ_[p] for the corresponding prime p.
```
instance (v : HeightOneSpectrum ℤ) : CompactSpace (v.adicCompletionIntegers ℚ) := by
…
```
instance Matrix.GeneralLinearGroup.(instance at line 148) line 148 source
def Matrix.GeneralLinearGroup.integralAdeles The integral adeles Ẑ = ∏ᵥ ℤᵥ as a subring of the finite adeles of ℚ: the adeles that are integral at every place line 154 source
Source documentation
The integral adeles Ẑ = ∏ᵥ ℤᵥ as a subring of the finite adeles of ℚ: the adeles that are integral at every place.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subring, IsDedekindDomain.FiniteAdeleRing, Int, Rat
Proof uses: Subring.mk, Subsemiring.mk, Submonoid.mk, Subsemigroup.mk, Set.ofPred, IsDedekindDomain.HeightOneSpectrum, Membership.mem, IsDedekindDomain.HeightOneSpectrum.adicCompletion, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers
38 instances, coercions and auxiliary constants left out.
```
def integralAdeles : Subring 𝔸ᶠ[ℤ, ℚ] where
…
```
lemma Matrix.GeneralLinearGroup.isOpen_integralAdeles line 162 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsOpen, IsDedekindDomain.FiniteAdeleRing, Int, Rat, Subring, Matrix.GeneralLinearGroup.integralAdeles
Proof uses: RestrictedProduct.isOpen_forall_mem, IsDedekindDomain.HeightOneSpectrum, IsDedekindDomain.HeightOneSpectrum.adicCompletion, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Valued.isOpen_valuationSubring, WithZero, Multiplicative
33 instances, coercions and auxiliary constants left out.
lemma Matrix.GeneralLinearGroup.isCompact_integralAdeles line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsCompact, IsDedekindDomain.FiniteAdeleRing, Int, Rat, Subring, Matrix.GeneralLinearGroup.integralAdeles
Proof uses: Eq, Set, Set.range, RestrictedProduct, IsDedekindDomain.HeightOneSpectrum, IsDedekindDomain.HeightOneSpectrum.adicCompletion, ValuationSubring, IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers, Filter.cofinite, RestrictedProduct.structureMap, Eq.mpr, Set.ofPred, Membership.mem, Filter.Eventually, id, congrArg, RestrictedProduct.range_structureMap, Eq.refl, Set.image, Set.univ and 7 more
47 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
def Matrix.GeneralLinearGroup.integralAdelicSubgroup GL n Ẑ inside GL n 𝔸ᶠ[ℤ, ℚ]: the matrices whose entries, and whose inverse's entries, are integral adeles line 181 source
Source documentation
GL n Ẑ inside GL n 𝔸ᶠ[ℤ, ℚ]: the matrices whose entries, and whose inverse's entries, are integral adeles. It is a compact open subgroup of G(𝔸_f) (isOpen_integralAdelicSubgroup, isCompact_integralAdelicSubgroup), as condition (b1) requires.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, Subgroup, Matrix.GeneralLinearGroup, IsDedekindDomain.FiniteAdeleRing, Int, Rat, Matrix
Proof uses: Subgroup.mk, Submonoid.mk, Subsemigroup.mk, Set.ofPred, And, Membership.mem, Subring, Matrix.GeneralLinearGroup.integralAdeles, Inv.inv, Set
44 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
def integralAdelicSubgroup : Subgroup (GL n 𝔸ᶠ[ℤ, ℚ]) where
…
```
lemma Matrix.GeneralLinearGroup.isOpen_integralAdelicSubgroup line 199 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, IsOpen, Matrix.GeneralLinearGroup, IsDedekindDomain.FiniteAdeleRing, Int, Rat, Matrix, Subgroup, Matrix.GeneralLinearGroup.integralAdelicSubgroup
Proof uses: Set.ofPred, Membership.mem, Subring, Matrix.GeneralLinearGroup.integralAdeles, Eq, Set, Set.iInter, Set.preimage, Set.ext, of_eq_true, Iff, Eq.trans, True, congrArg, forall_congr, iff_self, Eq.mpr, id, isOpen_iInter_of_finite, IsOpen.preimage and 8 more
52 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Matrix.GeneralLinearGroup.isCompact_integralAdelicSubgroup line 210 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, IsCompact, Matrix.GeneralLinearGroup, IsDedekindDomain.FiniteAdeleRing, Int, Rat, Matrix, Subgroup, Matrix.GeneralLinearGroup.integralAdelicSubgroup
Proof uses: Set, Set.ofPred, Membership.mem, Subring, Matrix.GeneralLinearGroup.integralAdeles, Eq, rfl, Eq.rec, Set.pi, Set.univ, isCompact_univ_pi, Matrix.GeneralLinearGroup.isCompact_integralAdeles, Eq.symm, Set.ext, Iff.intro, trivial, Prod, MulOpposite, Set.image, Units and 36 more
94 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.