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I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the cusps so that the natural projection maps obtained by forgetting the level structure are log-etale (at least away from primes dividing the order of your level structure).

I can have some rough intuition about how this happens over a field of characteristic zero, but not integrally. Can anybody explain this or give me a reference for this structure?

Additionally, has anybody worked out the appropriate integral ring of modular forms with logarithmic structure in some cases, similar to the Deligne-Tate calculation of modular forms over Z?

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2 Answers 2

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I thing Kato's log purity theorem gives you this. See, for instance, Theorem B in Mochizuki's "Extending Families of Curves over Log Regular Schemes." I think all you need is that the cusps form a normal crossings divisor on X(1) [if you're worried about X(1) being a stack rather than a scheme, you can start with a bit of extra level structure coprime to the primes you're interested in] and then your map Y(N) -> Y(1) is tamely ramified, which tells you that the normalization X(N) of X(1) in Y(N) carries a canonical log-structure in which the map X(N) -> X(1) is log-etale.

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If you're working away from the primes dividing the level, your curves have semi-stable reduction, and have canonical log-smooth log structures. For any pair (X,D), where X is smooth and D is a divisor with normal crossings, there is a log structure given by the set of functions in Ox that are invertible away from D. In your case, I think you take X to be the universal curve, and D to be the divisor at infinity. Forgetting a coprime level structure yields a map with vanishing log-cotangent complex.

References (may not have your precise statement):

  • F. Kato. Log smooth deformation theory
  • M. Olsson "Universal log structures on semistable varieties"

Olsson has some other papers that might be useful. He takes them off his web page when they get published, but sometimes you can find them with Google Scholar.

Edit: I haven't seen any work on the log-canonical rings of modular curves, but I don't really work in that area. You should allow poles of order n/2 for weight n forms, so for level 1, you get extra stuff like E14/Delta.

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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). –  Tyler Lawson Oct 15 '09 at 16:00
    
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. –  S. Carnahan Oct 15 '09 at 16:37
    
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. –  Tyler Lawson Oct 15 '09 at 18:09
    
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. –  S. Carnahan Oct 15 '09 at 20:37
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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. –  Tyler Lawson Sep 23 '10 at 1:27

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