Write a bit about modular form

[user202729]

Another guest chapter, I guess.

There's a few things to look out for:

• Now Asymptote compilation always input macros.tex, which results in \Re appears as \operatorname{Re} instead of ℜ. I think this is more convenient, but it leads to each Asymptote file takes like 0.02 seconds longer. I think this is negligible (remember to not add heavy packages to macros.tex then, otherwise total compilation time for all the Asymptote figures will blow up)
• I move some heavy parts from macros.tex to preamble.tex. Otherwise Asymptote compilation time would be like 1 second longer per figure.

Feel free to revert these (but if you do then change \Re/Im to \operatorname{Re/Im} in the plot).

If it's accepted I guess some \mathscr{P} can be changed to \PP in the category theory chapter (Cohen forcing), and maybe some \operatorname can be changed to \opname elsewhere.

• I feel I didn't do modular form enough justice, I guess. https://www.quantamagazine.org/behold-modular-forms-the-fifth-fundamental-operation-of-math-20230921/ has a good introduction (except the relation to lattice which I can't really find in any source…), I can't really write that well.

• there are some parts of commentary that I comment out… it is not your opinion though.

• I guess I could add "for evanchen's actual writing, see 18-786 number theory II course notes". The 10 pages I wrote is worth about 3 pages of course notes, I guess.

TODO:

• explain relation with elliptic curve
• explain modular functions over subgroups of SL₂(ℤ) (which corresponds to meromorphic functions on the space of marked lattice)
• relation to Θ function etc.? (I haven't learnt this one yet)

[vEnhance]

Is there a plan to add more chapters in the modular forms part?

If not (which is fine), then I think it might make more sense to not have a separate part just for this and merge it into the previous complex analysis section.

If there are more chapters planned, then I might rather wait until some more of those are written out to merge, so the chapter numbers don't keep jiggling around too much.

I'm a little reluctant to add input macros.tex because it makes the Asymptote code more brittle and less portable. I think it's not worth the extra dependency link if the only cost to not having it is having \operatorname{Re} in a few places.

[user202729]

We wait then.

[msm-code]

Oh, cool.

I'm currently (re)reading Napkin to brush up my number theory (and prepare for more advanced literature), and modular forms are on my roadmap soon. Happy to see this PR.

I don't know enough to say anything substantial about the content, but when adding a new part you probably also want to add it here: https://github.com/vEnhance/napkin/blob/main/tex/frontmatter/digraph.tex. I assume complex analysis is a prerequisite.

Btw, I chuckled at the unintentional humor of

Fundamentally, a modular form is something very simple: a meromorphic homogeneous function on the space of lattices of C

(I know that's in comparison to the regular definition, but it made me think of the monad joke)