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?



Logarithmic structures on moduli of elliptic curves over Z
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. 


