most recent 30 from http://mathoverflow.net 2013-05-19T05:30:57Z http://mathoverflow.net/feeds/question/592 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/592/logarithmic-structures-on-moduli-of-elliptic-curves-over-z Tyler Lawson 2009-10-15T12:26:34Z 2009-10-16T02:36:17Z

<p>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).</p> <p>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?</p> <p>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?</p>

http://mathoverflow.net/questions/592/logarithmic-structures-on-moduli-of-elliptic-curves-over-z/601#601 S. Carnahan 2009-10-15T15:13:42Z 2009-10-16T02:36:17Z

<p>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 O_x 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.</p> <p>References (may not have your precise statement): </p> <ul> <li>F. Kato. Log smooth deformation theory</li> <li>M. Olsson "Universal log structures on semistable varieties"</li> </ul> <p>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.</p> <p><b>Edit:</b> 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.</p>

http://mathoverflow.net/questions/592/logarithmic-structures-on-moduli-of-elliptic-curves-over-z/693#693 JSE 2009-10-16T01:47:30Z 2009-10-16T01:47:30Z

<p>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. </p>