Modular forms

This file defines modular forms and proves some basic properties about them.

We begin by defining modular forms and cusp forms as extension of SlashInvariantForms then we define the space of modular forms, cusp forms and prove that the product of two modular forms is a modular form.

structure ModularForm (Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)) (k : ℤ) extends SlashInvariantForm : Type

• toFun : UpperHalfPlane → ℂ
• slash_action_eq' : ∀ (γ : { x // x ∈ Γ }), SlashAction.map ℂ k γ s.toFun = s.toFun
• holo' : MDifferentiable (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ↑s.toSlashInvariantForm
• bdd_at_infty' : ∀ (A : Matrix.SpecialLinearGroup (Fin 2) ℤ), UpperHalfPlane.IsBoundedAtImInfty (SlashAction.map ℂ k A ↑s.toSlashInvariantForm)

These are SlashInvariantForm's that are holomophic and bounded at infinity.

structure CuspForm (Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)) (k : ℤ) extends SlashInvariantForm : Type

• toFun : UpperHalfPlane → ℂ
• slash_action_eq' : ∀ (γ : { x // x ∈ Γ }), SlashAction.map ℂ k γ s.toFun = s.toFun
• holo' : MDifferentiable (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ↑s.toSlashInvariantForm
• zero_at_infty' : ∀ (A : Matrix.SpecialLinearGroup (Fin 2) ℤ), UpperHalfPlane.IsZeroAtImInfty (SlashAction.map ℂ k A ↑s.toSlashInvariantForm)

These are SlashInvariantForms that are holomophic and zero at infinity.

class ModularFormClass (F : Type u_2) (Γ : outParam (Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ))) (k : outParam ℤ) extends SlashInvariantFormClass : Type u_2

• coe : F → UpperHalfPlane → ℂ
• coe_injective' : Function.Injective FunLike.coe
• slash_action_eq : ∀ (f : F) (γ : { x // x ∈ Γ }), SlashAction.map ℂ k γ ↑f = ↑f
• holo : ∀ (f : F), MDifferentiable (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ↑f
• bdd_at_infty : ∀ (f : F) (A : Matrix.SpecialLinearGroup (Fin 2) ℤ), UpperHalfPlane.IsBoundedAtImInfty (SlashAction.map ℂ k A ↑f)

ModularFormClass F Γ k says that F is a type of bundled functions that extend SlashInvariantFormClass by requiring that the functions be holomorphic and bounded at infinity.

class CuspFormClass (F : Type u_2) (Γ : outParam (Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ))) (k : outParam ℤ) extends SlashInvariantFormClass : Type u_2

• coe : F → UpperHalfPlane → ℂ
• coe_injective' : Function.Injective FunLike.coe
• slash_action_eq : ∀ (f : F) (γ : { x // x ∈ Γ }), SlashAction.map ℂ k γ ↑f = ↑f
• holo : ∀ (f : F), MDifferentiable (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ↑f
• zero_at_infty : ∀ (f : F) (A : Matrix.SpecialLinearGroup (Fin 2) ℤ), UpperHalfPlane.IsZeroAtImInfty (SlashAction.map ℂ k A ↑f)

CuspFormClass F Γ k says that F is a type of bundled functions that extend SlashInvariantFormClass by requiring that the functions be holomorphic and zero at infinity.

instance ModularFormClass.modularForm (Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)) (k : ℤ) : ModularFormClass (ModularForm Γ k) Γ k

instance CuspFormClass.cuspForm (Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)) (k : ℤ) : CuspFormClass (CuspForm Γ k) Γ k

theorem ModularForm.toFun_eq_coe {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) : f.toFun = ↑f

@[simp]

theorem ModularForm.toSlashInvariantForm_coe {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) : ↑f.toSlashInvariantForm = ↑f

theorem CuspForm.toFun_eq_coe {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {f : CuspForm Γ k} : f.toFun = ↑f

@[simp]

theorem CuspForm.toSlashInvariantForm_coe {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) : ↑f.toSlashInvariantForm = ↑f

theorem ModularForm.ext {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {f : ModularForm Γ k} {g : ModularForm Γ k} (h : ∀ (x : UpperHalfPlane), ↑f x = ↑g x) : f = g

theorem CuspForm.ext {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {f : CuspForm Γ k} {g : CuspForm Γ k} (h : ∀ (x : UpperHalfPlane), ↑f x = ↑g x) : f = g

def ModularForm.copy {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) (f' : UpperHalfPlane → ℂ) (h : f' = ↑f) : ModularForm Γ k

Copy of a ModularForm with a new toFun equal to the old one. Useful to fix definitional equalities.

def CuspForm.copy {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) (f' : UpperHalfPlane → ℂ) (h : f' = ↑f) : CuspForm Γ k

Copy of a CuspForm with a new toFun equal to the old one. Useful to fix definitional equalities.

instance ModularForm.add {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Add (ModularForm Γ k)

@[simp]

theorem ModularForm.coe_add {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) (g : ModularForm Γ k) : ↑(f + g) = ↑f + ↑g

@[simp]

theorem ModularForm.add_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) (g : ModularForm Γ k) (z : UpperHalfPlane) : ↑(f + g) z = ↑f z + ↑g z

instance ModularForm.instZero {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Zero (ModularForm Γ k)

@[simp]

theorem ModularForm.coe_zero {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : ↑0 = 0

@[simp]

theorem ModularForm.zero_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (z : UpperHalfPlane) : ↑0 z = 0

instance ModularForm.instSMul {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {α : Type u_2} [SMul α ℂ] [IsScalarTower α ℂ ℂ] : SMul α (ModularForm Γ k)

@[simp]

theorem ModularForm.coe_smul {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {α : Type u_2} [SMul α ℂ] [IsScalarTower α ℂ ℂ] (f : ModularForm Γ k) (n : α) : ↑(n • f) = n • ↑f

@[simp]

theorem ModularForm.smul_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {α : Type u_2} [SMul α ℂ] [IsScalarTower α ℂ ℂ] (f : ModularForm Γ k) (n : α) (z : UpperHalfPlane) : ↑(n • f) z = n • ↑f z

instance ModularForm.instNeg {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Neg (ModularForm Γ k)

@[simp]

theorem ModularForm.coe_neg {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) : ↑(-f) = -↑f

@[simp]

theorem ModularForm.neg_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) (z : UpperHalfPlane) : ↑(-f) z = -↑f z

instance ModularForm.instSub {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Sub (ModularForm Γ k)

@[simp]

theorem ModularForm.coe_sub {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) (g : ModularForm Γ k) : ↑(f - g) = ↑f - ↑g

@[simp]

theorem ModularForm.sub_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) (g : ModularForm Γ k) (z : UpperHalfPlane) : ↑(f - g) z = ↑f z - ↑g z

instance ModularForm.instAddCommGroupModularForm {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : AddCommGroup (ModularForm Γ k)

@[simp]

theorem ModularForm.coeHom_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : ModularForm Γ k) (a : UpperHalfPlane) : ↑ModularForm.coeHom f a = ↑f a

def ModularForm.coeHom {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : ModularForm Γ k →+ UpperHalfPlane → ℂ

Additive coercion from ModularForm to ℍ → ℂ.

instance ModularForm.instModuleComplexModularFormInstSemiringComplexToAddCommMonoidInstAddCommGroupModularForm {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Module ℂ (ModularForm Γ k)

instance ModularForm.instInhabitedModularForm {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Inhabited (ModularForm Γ k)

def ModularForm.mul {k_1 : ℤ} {k_2 : ℤ} {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} (f : ModularForm Γ k_1) (g : ModularForm Γ k_2) : ModularForm Γ (k_1 + k_2)

The modular form of weight k_1 + k_2 given by the product of two modular forms of weights k_1 and k_2.

@[simp]

theorem ModularForm.mul_coe {k_1 : ℤ} {k_2 : ℤ} {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} (f : ModularForm Γ k_1) (g : ModularForm Γ k_2) : ↑(ModularForm.mul f g) = ↑f * ↑g

instance ModularForm.instOneModularFormOfNatIntInstOfNatInt {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} : One (ModularForm Γ 0)

@[simp]

theorem ModularForm.one_coe_eq_one {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} : ↑1 = 1

instance CuspForm.hasAdd {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Add (CuspForm Γ k)

@[simp]

theorem CuspForm.coe_add {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) (g : CuspForm Γ k) : ↑(f + g) = ↑f + ↑g

@[simp]

theorem CuspForm.add_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) (g : CuspForm Γ k) (z : UpperHalfPlane) : ↑(f + g) z = ↑f z + ↑g z

instance CuspForm.instZero {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Zero (CuspForm Γ k)

@[simp]

theorem CuspForm.coe_zero {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : ↑0 = 0

@[simp]

theorem CuspForm.zero_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (z : UpperHalfPlane) : ↑0 z = 0

instance CuspForm.instSMul {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {α : Type u_2} [SMul α ℂ] [IsScalarTower α ℂ ℂ] : SMul α (CuspForm Γ k)

@[simp]

theorem CuspForm.coe_smul {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {α : Type u_2} [SMul α ℂ] [IsScalarTower α ℂ ℂ] (f : CuspForm Γ k) (n : α) : ↑(n • f) = n • ↑f

@[simp]

theorem CuspForm.smul_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} {α : Type u_2} [SMul α ℂ] [IsScalarTower α ℂ ℂ] (f : CuspForm Γ k) (n : α) {z : UpperHalfPlane} : ↑(n • f) z = n • ↑f z

instance CuspForm.instNeg {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Neg (CuspForm Γ k)

@[simp]

theorem CuspForm.coe_neg {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) : ↑(-f) = -↑f

@[simp]

theorem CuspForm.neg_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) (z : UpperHalfPlane) : ↑(-f) z = -↑f z

instance CuspForm.instSub {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Sub (CuspForm Γ k)

@[simp]

theorem CuspForm.coe_sub {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) (g : CuspForm Γ k) : ↑(f - g) = ↑f - ↑g

@[simp]

theorem CuspForm.sub_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) (g : CuspForm Γ k) (z : UpperHalfPlane) : ↑(f - g) z = ↑f z - ↑g z

instance CuspForm.instAddCommGroupCuspForm {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : AddCommGroup (CuspForm Γ k)

@[simp]

theorem CuspForm.coeHom_apply {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} (f : CuspForm Γ k) (a : UpperHalfPlane) : ↑CuspForm.coeHom f a = ↑f a

def CuspForm.coeHom {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : CuspForm Γ k →+ UpperHalfPlane → ℂ

Additive coercion from CuspForm to ℍ → ℂ.

instance CuspForm.instModuleComplexCuspFormInstSemiringComplexToAddCommMonoidInstAddCommGroupCuspForm {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Module ℂ (CuspForm Γ k)

instance CuspForm.instInhabitedCuspForm {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} : Inhabited (CuspForm Γ k)

instance CuspForm.instModularFormClass {F : Type u_1} {Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)} {k : ℤ} [CuspFormClass F Γ k] : ModularFormClass F Γ k