Timeline for How do you recover the structure of the upper half plane from its description as a coset space?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 30, 2010 at 4:59 | vote | accept | Qiaochu Yuan | ||
| Mar 30, 2010 at 4:09 | answer | added | S. Carnahan♦ | timeline score: 4 | |
| Mar 29, 2010 at 20:27 | comment | added | Kevin Buzzard | @Mariano: it's not obvious that SL_2(R)/SO_2(R) has a complex structure, and it's not obvious that SL_2(Z)\SL_2(R)/SO_2(R) has a complex structure, but I'm sure you can see that one has a complex structure iff the other one does. He asked about the first but I mentioned the second because I am pretty sure he knows SL_2(Z)\H is a moduli space and I'm less sure if he knows that H is (it's also another moduli space but it's of elliptic curves plus more structure). | |
| Mar 29, 2010 at 19:06 | comment | added | Mariano Suárez-Álvarez | Hmmm. What does SL_2(Z) \ H have to do with Qiaochu's question? | |
| Mar 29, 2010 at 18:59 | comment | added | Idoneal | I get your point but I have a confusion now. It seems to me that over the complex numbers things are a little easier. Namely, an elliptic curve is just C modulo a lattice and isomorphim classes of elliptic curves are the same things as classes of lattices up to homothety from which SL_2(Z)\H appears rather naturally. I do agree, however, that in general, things will not be so easy. | |
| Mar 29, 2010 at 18:28 | comment | added | Kevin Buzzard | ...and the point is that you want M-->S to be a holomorphic family of elliptic curves iff the induced map from the complex manifold S to the set of iso classes of ell curves (sending s to the fibre above s) is holomorphic. This gives you the structure of a complex manifold on SL_2(Z) \ H. But now one has to prove that it's the same one as the one coming from the upper half plane, and this genuinely needs proof. | |
| Mar 29, 2010 at 18:27 | comment | added | Kevin Buzzard | Sorry Idoneal, I didn't explain myself well. I claim that there is a canonical bijection between the set of isomorphism classes of elliptic curves over the complexes, and SL_2(Z) \ H. If you're prepared to believe that H is a complex manifold, then this set becomes a complex manifold. However if you didn't know H existed, and just had a set of isomorphism classes of elliptic curves, however would you put a complex structure on it? It can be done! You need to consider holomorphic families of ell curves (i.e. maps M-->S of cx mfds whose fibres are ell curves)... | |
| Mar 29, 2010 at 18:21 | comment | added | Idoneal | "t's not hard to check that the points of SL_2(Z) \ H parametrise elliptic curves over the complexes, but why should such a set be a complex manifold?" I am not sure if I understood you correctly. What's wrong with the usual proof (say, in Silverman's Advanced Topics in Elliptic Curves") that shows that after giving the appropriate charts and adding an extra point at infinity (one needs to be careful about the elliptic points) this is biholomorphic to the Riemann sphere? | |
| Mar 29, 2010 at 18:18 | answer | added | José Figueroa-O'Farrill | timeline score: 4 | |
| Mar 29, 2010 at 17:49 | comment | added | Kevin Buzzard | @Qiaochu: that's a highly non-dumb question! It comes up in the theory of moduli spaces. It's not hard to check that the points of SL_2(Z) \ H parametrise elliptic curves over the complexes, but why should such a set be a complex manifold? I think there's a very fancy answer to do with variation of Hodge structures but I don't think I ever understood the details of that point of view well enough to be able to explain them :-( . I somehow feel that Deligne's axioms for a Shimura variety should somehow help, but on some level I've never understood these either :-( | |
| Mar 29, 2010 at 17:09 | answer | added | Charlie Frohman | timeline score: 7 | |
| Mar 29, 2010 at 16:59 | history | asked | Qiaochu Yuan | CC BY-SA 2.5 |