Timeline for On the algberaicity of the universal elliptic curve associated to a torsion free subgroup
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| May 31, 2012 at 21:00 | vote | accept | Hugo Chapdelaine | ||
| May 31, 2012 at 21:00 | history | bounty ended | Hugo Chapdelaine | ||
| May 30, 2012 at 3:12 | comment | added | Keerthi Madapusi | I guess for this kind of argument to work, you really need to compactify everything, which is a pain. This is done in Brian Conrad's very meticulous, but still (and probably forever) unpublished notes. See www.math.leidenuniv.nl/~edix/public_html_rennes/brian.ps. The result is Corollary 3.3.5.2. | |
| May 30, 2012 at 2:58 | comment | added | Keerthi Madapusi | Okay, I now see the problem with my argument. Even if this sheaf is analytically a line bundle, it need not be one algebraically. | |
| May 30, 2012 at 2:12 | comment | added | Keerthi Madapusi | Hi Ben-By 'family of elliptic curves' I meant a relative elliptic curve in the arithmetic geometer's sense. In particular, it admits a global identity section that Donu calls $\sigma$ below. After that the argument is basically what he sketches, except that I had in mind a semi-continuity and base-change argument to show that $f_*\mathcal{O}(3\sigma)$ is trivial. I'm not sure if these results are true in the analytic category, but my sense was that this is where they in fact originate. | |
| May 29, 2012 at 18:22 | answer | added | Misha | timeline score: 3 | |
| May 29, 2012 at 17:43 | answer | added | Donu Arapura | timeline score: 3 | |
| May 29, 2012 at 14:44 | history | bounty started | Hugo Chapdelaine | ||
| May 28, 2012 at 13:10 | comment | added | Hugo Chapdelaine | Good example Ben. One possibility to show algebraicity (which I find a bit complicated) would be to construct a very ample line bundle of theta Poincare series and to show that it lift to a suitable compactification of $Y_{\Gamma}$. But such a program looks more complicated that the original question... | |
| May 27, 2012 at 21:49 | comment | added | Ben Wieland | Keerthi, on the contrary, the Hopf surface is the easiest example of a non-algebraic surface. It is an analytic surface $(\mathbb C^2-0) / q^{\mathbb Z}$ with evident map to $P^1=(\mathbb C^2-0)/\mathbb C^*$ exhibiting it as a bundle with fiber all elliptic curves isomorphic to $\mathbb C^\* / q^{\mathbb Z}$, but its function field is one dimensional. en.wikipedia.org/wiki/Hopf_surface | |
| May 27, 2012 at 10:47 | comment | added | Hugo Chapdelaine | @Keerthi, I don't quite see how to do it. For example if you take the usual Weierstrass model then you need to exclude the $j$-invariants $1728$ and $0$. If we had a universal elliptic curve over $H/SL_2(\mathbf{Z})$ then we could just take the pullback of the projection $\pi:H/\Gamma\rightarrow H/SL_2(\mathbf{Z})$ but we don't have this.. | |
| May 27, 2012 at 10:27 | comment | added | Hugo Chapdelaine | I see, so I'll try to sketch the argument following your comment | |
| May 26, 2012 at 16:22 | comment | added | Keerthi Madapusi | In general, if $X$ is an algebraic variety over $\mathbb{C}$ and $E\to X$ is an analytic family of elliptic curves, then $E$ is automatically algebraic, and in fact projective over $X$. In particular, if $X$ is quasi-projective, then so is $E$. The point is that the Weierstrass embedding of an elliptic curve into $\mathbb{P}^2$ works in families as well. | |
| May 26, 2012 at 11:07 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
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| May 24, 2012 at 20:15 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
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| May 24, 2012 at 19:52 | comment | added | Hugo Chapdelaine | So I replaced family of elliptic curves for family of curves! | |
| May 24, 2012 at 19:51 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
deleted 456 characters in body; added 40 characters in body
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| May 24, 2012 at 19:34 | comment | added | Hugo Chapdelaine | You are right I was a bit careless in the set up | |
| May 24, 2012 at 18:24 | comment | added | Kevin Buzzard | I should of course say that my comments do not pertain to the question (which restricts to the torsion-free case); they just pertain to the background stuff. | |
| May 24, 2012 at 18:22 | comment | added | Kevin Buzzard | You have to be careful here. I am not sure that you have written down a family of elliptic curves in general. For example if $-1\in\Gamma$ then you seem to be identifying $(\tau,x)$ with $(\tau,-x)$. There may be similar problems with elliptic points even if $-1\not\in\Gamma$ (for example if $\Gamma=\Gamma_1(3)$). The problem is that your moduli problem is not representable in general, and the way you've set it up I think that in the curve above $\tau$ is the torus you think it is, modulo e.g. its automorphism group, which from what you write is I think not what you want to happen. | |
| May 24, 2012 at 16:53 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
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| May 24, 2012 at 16:32 | history | asked | Hugo Chapdelaine | CC BY-SA 3.0 |