Timeline for Arguing that weakly holomorphic modular forms give rise to Katz modular forms
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| Oct 31, 2017 at 14:50 | comment | added | nfdc23 | In my first comment I sketched how to prove GAGA for DM stacks; I don't know a reference. A classical form meromorphic at the cusps is a global section of a twist by a positive power of the inverse ideal sheaf of the cuspidal substack against the sheaf without allowing poles, and the latter is built as in the algebro-geometric case from the universal generalized elliptic curve over the moduli stack. You can define "analytifications" by descent from higher-level scheme cases; the notation $[\overline{\mathcal{H}}/\Gamma]$ makes no sense since $\overline{\mathcal{H}}$ isn't a complex manifold. | |
| Oct 30, 2017 at 22:21 | comment | added | stupid_question_bot | @nfdc23 Right, so my questions are: (1) Where can I find a statement of GAGA for Deligne-Mumford stacks (which ideally doesn't use infinity-stack terminology), and (2) Given a classical modular form $f$, meromorphic at all cusps, how does one define a section of the relevant line bundle on the proper stack $[\overline{\mathcal{H}}/\Gamma]$? (I think I can do it on the open substack $[\mathcal{H}/\Gamma]$, but I'm still not sure how to do it near the cusps. | |
| Oct 30, 2017 at 6:03 | comment | added | nfdc23 | OK, I didn't read your entire question (it was too long for me), so I was just guessing that you were starting from the purely analytic side, which I now see you weren't doing. Being "weakly holomorphic" with pole-order below some bound at the cusps corresponds to a global section of the evident line bundle on the proper stack by imposing some twist of the usual line bundle by a positive power of the inverse ideal sheaf of the cusps, so you can just applying GAGA to conclude without having to say anything about "Tate curves" or $q$-expansions. | |
| Oct 30, 2017 at 5:40 | comment | added | stupid_question_bot | @nfdc23 If $F$ is a Katz modular form, which determines a classical modular form $f$, then evaluating at the Tate curve over $\mathbb{C}((q^{1/n}))$ gives you an element of $\mathbb{C}((q^{1/n}))$ which I believe is precisely the $q$-expansion of $f$ at some cusp, depending on the level structure one chooses - but elements of $\mathbb{C}((q^{1/n}))$ are laurent series, and all level structures are defined over $\mathbb{C}((q^{1/n}))$ so doesn't that mean $f$ can't have any essential singularities at the cusps? | |
| Oct 30, 2017 at 5:31 | comment | added | nfdc23 | Working in the purely analytic setting as you are, how are you inferring meromorphicity rather than the possibility of essential singularities? Saying "evaluate at the Tate curve" doesn't make the issue of essential singularities disappear. Also, since there is Chow's Lemma for separated Artin stacks of finite type, to prove a GAGA-type theorem (you only need DM stacks, which is technically simpler) you can bootstrap from the cases of proper algebraic spaces and/or schemes exactly as Grothendieck does from the projective to the proper case for schemes in Expose XII of SGA1. | |
| Oct 30, 2017 at 4:25 | history | edited | stupid_question_bot | CC BY-SA 3.0 |
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| Oct 30, 2017 at 4:16 | history | asked | stupid_question_bot | CC BY-SA 3.0 |