feat: bounds for modular forms of non-positive weights

[CBirkbeck]

We use the Maximum modulus principle to show that modular forms of non-positive weights are bounded by its value of some element of the upper half plane imaginary part at least 1/2.

---

[github-actions[bot]]

PR summary 7d5154c184

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.NumberTheory.ModularForms.LevelOne	1646

---

Declarations diff

+ SlashInvariantForm.exists_one_half_le_im_and_norm_le + SlashInvariantForm.wt_eq_zero_of_eq_const + coe + coe_apply_complex + denom_S + denom_one + det_coe + exists_one_half_le_im_smul + exists_one_half_le_im_smul_and_norm_denom_le + three_le_four_mul_im_sq_of_mem_fd -++- im_smul_eq_div_normSq

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.

---

Decrease in tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type
1552	-1	erw

Current commit 7d5154c184 Reference commit 66df24a8e2

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).

[loefflerd]

Following some discussion with @CBirkbeck, I've pushed some changes to this branch which avoid making subgroups of SL2Z into subtypes, which gets rid of some unnecessary coercions and API lemmas.

[mergify[bot]]

⚠️ The sha of the head commit of this PR conflicts with #18385. Mergify cannot evaluate rules on this PR. ⚠️

[CBirkbeck]

I made #18385 to make the reviewing easier. That PR contains all the q.o.l improvements, this has the new results.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• ~~leanprover-community/mathlib4#18385~~ By Dependent Issues (🤖). Happy coding!

[loefflerd]

This had a merge-conflict because of #18385. I merged it with the latest master and sorted out the conflicts, ready for review again.

[CBirkbeck]

This had a merge-conflict because of #18385. I merged it with the latest master and sorted out the conflicts, ready for review again.

thanks for doing this and again below!

[loefflerd]

No problem! By the way, I realised while fixing the conflicts that we have two distinct namespaces ModularGroup and UpperHalfPlane.ModularGroup. Is this deliberate or should they be unified? In any case denom_one as presently formulated shouldn't be in the ModularGroup namespace, since the 1 is getting elaborated as 1 : ↥GL(2, ℝ)⁺ rather than as 1 : SL(2, ℤ).

[CBirkbeck]

No problem! By the way, I realised while fixing the conflicts that we have two distinct namespaces ModularGroup and UpperHalfPlane.ModularGroup. Is this deliberate or should they be unified? In any case denom_one as presently formulated shouldn't be in the ModularGroup namespace, since the 1 is getting elaborated as 1 : ↥GL(2, ℝ)⁺ rather than as 1 : SL(2, ℤ).

No that was a mistake. I've removed it now and move denom_one.

[loefflerd]

(I can push fixes with all these changes to your PR, just let me know if that's OK with you.)

[CBirkbeck]

(I can push fixes with all these changes to your PR, just let me know if that's OK with you.)

Oh I was just doing this now. Go ahead and I'll double check there is nothing missing.

[loefflerd]

I have pushed fixes, I believe everything outside the new file LevelOne is now OK. I have not looked in detail at the new file, other than fixing knock-on effects of the changes I made elsewhere. Looking at it now.

[loefflerd]

I think the results like modular_S_smul and those near it should maybe also move down

That doesn't work because it's used in exists_SL2_smul_eq_of_apply_zero_one_ne_zero (which is definitely not a ModularGroup lemma).

[github-actions[bot]]

🚀 Pull request has been placed on the maintainer queue by loefflerd.

[riccardobrasca]

Thanks!

bors d+

[mathlib-bors[bot]]

:v: CBirkbeck can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

[CBirkbeck]

bors r+

[mathlib-bors[bot]]

Pull request successfully merged into master.

Build succeeded:

• Build
• Lint style