Timeline for Generators of the graded ring of modular forms
Current License: CC BY-SA 3.0
16 events
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| Jul 21, 2015 at 6:44 | comment | added | David Loeffler | Great! That already gets us down to 4 for $\Gamma_0(N)$, since it always contains $-1$ (so there are no nonzero forms of weight 5) and has at least two cusps for $N > 1$. | |
| Jul 20, 2015 at 2:24 | comment | added | Aaron Landesman | Let's say you want to reduce the bound from 6 to 5. By Theorem 4.1.1 in Voight and Zurieck-Brown (combined with 8.3.1), a congruence subgroup with no order 3 elliptic points is generated in weight at most 5 if and only if it has more than 1 cusp. One might further be able to reduce the bound from 5 to 4, as suggested in Question 8.4 of Landesman Ruhm Zhang by generalizing Reid's analysis on page 35 of homepages.warwick.ac.uk/~masda/Reprint/hyperplane.pdf to log divisors. | |
| Jul 20, 2015 at 0:58 | comment | added | David Loeffler | Can one improve on the bound 6? I ran a computation for $\Gamma_0(N)$ for $N \le 300$ or something, and convinced myself that if there are no order 3 elliptic points then weight 4 is sufficient (and if there are no elliptic points at all, then weight 2 suffices as long as $N$ is not prime). | |
| Jul 17, 2015 at 0:28 | history | edited | Aaron Landesman | CC BY-SA 3.0 |
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| Jul 12, 2015 at 23:09 | history | edited | Aaron Landesman | CC BY-SA 3.0 |
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| Jul 12, 2015 at 22:57 | comment | added | John Voight | Well, the result over $\mathbb{Z}[1/6N]$ implies the result over $\mathbb{Q}$ and therefore any field of characteristic $0$, so in particular it answers the original question (for the subring in even weight). More generally, one can apply flat base change; but I don't know how far we want to get into this in these comments. | |
| Jul 12, 2015 at 20:35 | vote | accept | David Loeffler | ||
| Jul 12, 2015 at 18:06 | comment | added | Aaron Landesman | I haven't read chapter 11 carefully, but it looks like that only applies to the base ring $\mathbb Z[1/6N].$ How do you extend it to arbitrary base rings? Are you saying that when you tensor up, the generator and relation degrees will be preserved? I know this works for fields, but I haven't thought through it for general commutative rings with unit. | |
| Jul 12, 2015 at 17:58 | comment | added | John Voight | You may also want to refer to DZB's Proposition 11.3.1, since extends Theorem 9.3.1 to more general base rings. JV | |
| Jul 12, 2015 at 17:58 | comment | added | Aaron Landesman | Right, that was unclear. I removed the word minimal. I probably should say something like "a set of generators of the ideal of relations" to be perfectly clear, but that should be understood. | |
| Jul 12, 2015 at 17:56 | history | edited | Aaron Landesman | CC BY-SA 3.0 |
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| Jul 12, 2015 at 17:56 | comment | added | John Voight | What are "minimal relations"? I think you just mean "relations", since you say "at most 12" anyway. JV | |
| Jul 12, 2015 at 17:55 | history | edited | Aaron Landesman | CC BY-SA 3.0 |
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| Jul 12, 2015 at 17:38 | history | edited | Aaron Landesman | CC BY-SA 3.0 |
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| Jul 12, 2015 at 17:15 | review | First posts | |||
| Jul 12, 2015 at 17:33 | |||||
| Jul 12, 2015 at 17:12 | history | answered | Aaron Landesman | CC BY-SA 3.0 |