Timeline for Logarithmic structures on moduli of elliptic curves over Z
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Sep 23, 2010 at 5:18 | comment | added | S. Carnahan♦ | Yes, that is correct - I think my understanding was rather weak when I wrote that answer. This is why weight 2 cusp forms correspond to global 1-forms. | |
| Sep 23, 2010 at 1:27 | comment | added | Tyler Lawson | Upon reviewing this, I should mention that I think that the standard rings of modular forms already incorporates the log structure. If you express complex-analytically the condition that the modular form $f(\tau)$ of weight $2k$ is holomorphic at $\infty$ in terms of $q = e^{2\pi i \tau}$, you find that it is equivalent to being of the form $g(q) dlog(q)^k$ for $g(q)$ holomorphic at the origin. | |
| Oct 16, 2009 at 2:36 | history | edited | S. Carnahan♦ | CC BY-SA 2.5 |
log-canonical rings
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| Oct 15, 2009 at 20:37 | comment | added | S. Carnahan♦ | This isn't really my milieu, but I don't think you get odd weight forms if your level structure admits the -1 automorphism. If you have odd weight forms, I guess you could say that geometric fibers of the divisor at infinity are representable, but it doesn't seem to be a log statement. | |
| Oct 15, 2009 at 18:09 | comment | added | Tyler Lawson | Actually, I was hoping that it was something more like the fact that every point on the divisor has an automorphism group of size at least 2 - but I'm unsure in general about trying to talk about divisors on stacks. | |
| Oct 15, 2009 at 16:37 | comment | added | S. Carnahan♦ | Google: no hits for "log spin structure" or "logarithmic spin structure". For notation, maybe \pi_*(\omega^1_{E/X})? The thing inside the parentheses implicitly involves the log structures. | |
| Oct 15, 2009 at 16:00 | comment | added | Tyler Lawson | Thanks. I'll look at the references, but there are a couple of things that immediately come to mind. I think of modular forms of even weight as sections of a power of the cotangent bundle, which has a square root w which is the line bundle of invariant 1-forms. How do you interpret modular forms of odd weight in the logarithmic sense? It seems odd to try to write w(D/2). | |
| Oct 15, 2009 at 15:13 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |