Timeline for Is the ring of meromorphic modular forms on a fine modular curve generated in degree 1?

6 events

when toggle format	what		by	license	comment
Oct 2, 2016 at 2:59	vote	accept	stupid_question_bot
Oct 2, 2016 at 2:15	answer	added	S. Carnahan♦		timeline score: 3
Sep 25, 2016 at 16:47	comment	added	Tyler Lawson		This doesn't work in the stack case because that you lose affineness; you also lose the property that "taking global sections" is an equivalence of categories.
Sep 25, 2016 at 16:45	comment	added	Tyler Lawson		I believe that this is true. If $X$ is the modular curve minus the cusps, there is a line bundle $\omega$ on it and the set of meromorphic modular forms of weight $k$ is the same as the set of (algebraic) sections of the line bundle $\omega^{\otimes k}$. However, $X$ is an affine scheme. This means that "taking sections" gives an equivalence between line bundles on $X$ and projective modules of rank 1 over $H^0(X;\cal{O})$, so that the group of forms of weight k is always the k-fold tensor of the group of sections of weight 1 over $H^0(X;{\cal O})$.
Sep 25, 2016 at 8:21	comment	added	David Loeffler		I've asked a couple of questions before about the graded ring of holomorphic modular forms: see math.stackexchange.com/questions/96395 (for $\Gamma$ sufficiently small) and mathoverflow.net/questions/66819 (for general $\Gamma$). This is a slightly different question, of course, but the methods might help for your question too.
Sep 24, 2016 at 23:48	history	asked	stupid_question_bot	CC BY-SA 3.0