[Merged by Bors] - feat(NumberTheory/ModularForms): define modular forms for any subgroup of GL(2, R)

[loefflerd]

This PR generalizes (some of) the modular forms files to allow the level to be any subgroup of GL(2, R), not necessarily
the image of a subgroup of SL(2, Z). Intended applications include defining Hecke operators.

---

• depends on: [Merged by Bors] - feat(LinearAlgebra/Projectivization): GL(V) action #26811
• depends on: [Merged by Bors] - feat(Compactification/OnePoint): classify GL2 elements by fixed points #28552
• depends on: [Merged by Bors] - feat(NumberTheory/ModularForms): Cusps of subgroups acting on upper half-plane #29327
• depends on: [Merged by Bors] - feat(NumberTheory/ModularForms): define boundedness / vanishing at a cusp #29560
• depends on: [Merged by Bors] - feat(NumberTheory/ModularForms): tweak def of slash-action #29804

[github-actions]

PR summary bf35bd39a0

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Basic	2471	2505	+34 (+1.38%)
Mathlib.NumberTheory.ModularForms.QExpansion	2475	2509	+34 (+1.37%)
Mathlib.NumberTheory.ModularForms.LevelOne	2488	2522	+34 (+1.37%)
Mathlib.NumberTheory.ModularForms.Basic	2452	2484	+32 (+1.31%)
Mathlib.NumberTheory.ModularForms.SlashInvariantForms	1853	1851	-2 (-0.11%)
Mathlib.NumberTheory.ModularForms.Petersson	1928	1926	-2 (-0.10%)

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.ModularForms.Petersson Mathlib.NumberTheory.ModularForms.SlashInvariantForms	-2
Mathlib.NumberTheory.ModularForms.Basic	32
3 files Mathlib.NumberTheory.ModularForms.EisensteinSeries.Basic Mathlib.NumberTheory.ModularForms.LevelOne Mathlib.NumberTheory.ModularForms.QExpansion	34

---

Declarations diff

+ CuspFormClass.zero_at_infty
+ CuspFormClass.zero_at_infty_slash
+ HasDetOne
+ HasDetPlusMinusOne
+ HasDetPlusMinusOne.abs_det
+ IsGLPos.instSMul
+ ModularFormClass.bdd_at_infty
+ ModularFormClass.bdd_at_infty_slash
+ coe_smulℝ
+ constℝ_apply
+ instModuleComplex
+ instModuleReal
+ instSMulℂ
+ instance (Γ : Subgroup (SL n R)) : HasDetOne (Γ.map toGL)
+ instance : Module ℝ (CuspForm Γ k)
+ instance : Module ℝ (ModularForm Γ k)
+ instance [HasDetOne Γ] : HasDetPlusMinusOne Γ := ⟨fun {g} hg ↦ by simp [HasDetOne.det_eq hg]⟩
+ instance [Γ.HasDetOne] : Module ℂ (CuspForm Γ k)
+ instance [Γ.HasDetOne] : Module ℂ (ModularForm Γ k)
+ instance [Γ.HasDetPlusMinusOne] : IntCast (ModularForm Γ 0)
+ instance [Γ.HasDetPlusMinusOne] : IntCast (SlashInvariantForm Γ 0)
+ instance [Γ.HasDetPlusMinusOne] : NatCast (ModularForm Γ 0)
+ instance [Γ.HasDetPlusMinusOne] : NatCast (SlashInvariantForm Γ 0)
+ instance [Γ.HasDetPlusMinusOne] : One (ModularForm Γ 0)
+ instance [Γ.HasDetPlusMinusOne] : One (SlashInvariantForm Γ 0)
+ instance {S : Type*} [CommRing S] [Algebra S R] (Γ : Subgroup (SL n S)) :
+ instance {Γ : Subgroup (GL (Fin 2) ℝ)} [h : Γ.IsArithmetic] : HasDetPlusMinusOne Γ := by
+ slash_action_eqn_SL''
+ smul_applyℝ
++ IsGLPos.coe_smul
++ IsGLPos.smul_apply
++ constℝ
++ instSMulℝ
++ instance (Γ : Subgroup (GL (Fin 2) ℝ)) [Γ.HasDetPlusMinusOne] :
++-- mul
- instance (Γ : Subgroup SL(2, ℤ)) : GradedMonoid.GMul (ModularForm Γ)
- instance (Γ : Subgroup SL(2, ℤ)) : GradedMonoid.GOne (ModularForm Γ)
- instance (Γ : Subgroup SL(2, ℤ)) : IntCast (ModularForm Γ 0)
- instance (Γ : Subgroup SL(2, ℤ)) : IntCast (SlashInvariantForm Γ 0)
- instance (Γ : Subgroup SL(2, ℤ)) : NatCast (ModularForm Γ 0)
- instance (Γ : Subgroup SL(2, ℤ)) : NatCast (SlashInvariantForm Γ 0)
- instance : Module ℂ (CuspForm Γ k)
- instance : Module ℂ (ModularForm Γ k)
- instance : Module ℂ (SlashInvariantForm Γ k)
- instance : One (ModularForm Γ 0)
- instance : One (SlashInvariantForm Γ 0)
-+- instSMul
-+-+ coe_intCast
-+-+ coe_natCast
--++ const
--++ one_coe_eq_one

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat(LinearAlgebra/Projectivization): GL(V) action #26811
• [Merged by Bors] - feat(Compactification/OnePoint): classify GL2 elements by fixed points #28552
• [Merged by Bors] - feat(NumberTheory/ModularForms): Cusps of subgroups acting on upper half-plane #29327
• [Merged by Bors] - feat(NumberTheory/ModularForms): define boundedness / vanishing at a cusp #29560
• [Merged by Bors] - feat(NumberTheory/ModularForms): tweak def of slash-action #29804
  By Dependent Issues (🤖). Happy coding!

[riccardobrasca]

@CBirkbeck can you please have a look?

[CBirkbeck]

why does this need the FiniteIndex condition now? We didn't have it before and the level isnt more general here right?

[loefflerd]

Ah, that's a subtle point, well spotted.

If you take a subgroup of Γ of SL2Z with infinite index, then the new definition of "cusp forms of level Γ" isn't the same as the old one. The new definition quantifies over the cusps of Γ, defined intrinsically as fixedpoints inside P^1(R) of parabolic elements of Γ; while the old definition just quantifies over P^1(Q) inside P^1(R), whatever Γ is.

Now, it's a theorem that if Γ has finite index, then its cusps are the same as those of SL2Z, namely P1(Q); hence the new definition is equivalent to the old one if Γ is finite-index. In particular, for a finite-index subgroup ∞ is always a cusp, which is what we need for this argument. On the other hand, if Γ has infinite index, then ∞ might not be a cusp; in the extreme case Γ = {1}, there aren't any cusps at all (because there are no parabolic elements).

So that's why we have picked up a FiniteIndex assumption here that wasn't there before.

[riccardobrasca]

Thanks!

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• CI Success