On the algberaicity of the universal elliptic curve associated to a torsion free subgroup - MathOverflow

On the algberaicity of the universal elliptic curve associated to a torsion free subgroup

2012-05-24T16:32:57Z

So let $\Gamma\subseteq SL_2(\mathbf{Z})$ be a finite index subgroup (not necessarily a congruence subgroup). Recall that we have an action of $SL_2(\mathbf{R})$ on $\mathbb{H}=\{z\in\mathbf{C}:\Im(z)>0\}$ by moebius transformations and therefore of $\Gamma$. If $\tau\in\mathbb{H}$ and $[a,b,c,d]=\gamma\in SL_2(\mathbf{Z})$ then we have an isomorphism of complex tori $$ \mathbf{C}/(\mathbf{Z}+\tau\mathbf{Z})\rightarrow \mathbf{C}/((a\tau+b)\mathbf{Z}+(c\tau+d)) \mathbf{Z}\rightarrow (\mathbf{C}/\mathbf{Z}+\gamma\tau\mathbf{Z}) \;\;\;\; (*) $$ where the first map is the identity and the second map is the multiplication by $(c\tau+d)^{-1}$. Let $$ \tilde{\mathcal{E}}_{\Gamma}=\{(\tau,x):\tau\in\mathbb{H},x\in \mathbf{C}/(\mathbf{Z}+\tau\mathbf{Z})\} $$ We have a natural left action of $\Gamma$ on $\tilde{\mathcal{E}}_{\Gamma}$ given by $$ \gamma(\tau,x)=(\gamma\tau,j(\gamma,\tau)^{-1}x), $$ which is just a reinterpretation of $(*)$. Here $j(\gamma,\tau)=c\tau+d$. We thus get the following family of curves (note that the fibers are not necessarily elliptic curves because of the presence of torsion in $\Gamma$ as K. Buzzard pointed out): $$ \pi_\Gamma:\Gamma\backslash\tilde{\mathcal{E}}_{\Gamma}=:\mathcal{E}_{\Gamma}\rightarrow Y_{\Gamma}:=\Gamma\backslash \mathbb{H} $$ In the case where $\Gamma$ is torsion free, we readily see that the fibers are elliptic curves and the the $\Gamma$ action is compatible with the addition on the $tori$ (this somehow justifies the terminology "universal elliptic curve" over $Y_{\Gamma}$).

In general one always has that $Y_{\Gamma}$ is a quasi-projective curve defined over $\mathbf{C}$ (in fact it is always possible to define this curve over $\overline{\mathbf{Q}}$, the algebraic closure of $\mathbf{Q}$).

[For example, when $\Gamma=\Gamma_0(N)$ one may look at the modular polynomial $f_N(x,y)\in\mathbf{Z}[x,y]$. Then using the modular interpretation one can show that there exists a (complex analytic) immersion of $Y_{\Gamma_0(N)}$ onto the plane curve $C_N: f_N(x,y)=0$. If I remember correctly, in the finite chart $\mathbf{C}^2$ (for $N$ large enough) the singularities of $C_N$ are nodes and thus one can blow-up these points (over $\mathbf{Q}$!). From this one can construct a complex analytic isomorphism between $Y_{\Gamma_0(N)}$ and the blow-up (which is quasi-projective curve defined over $\mathbf{Q}$.]

So here are 2 natural questions.

Q1: Is $\mathcal{E}_{\Gamma}$ quasi-projective (at least when $\Gamma$ is torsion free)?

Q2: If the answer is yes, then what is the cleanest (and if possible most transparent) way of showing that $\mathcal{E}_{\Gamma}$ is quasi-projective?

So for the second question, the thing that I have in mind would be to 1) construct some complex analytic immersion $\pi:\mathcal{E}_{\Gamma}\rightarrow Z$, where $Z$ is a quasi-projective surface and 2) performing a sequence of blow-ups on $Z$ I would try to construct an embedding.

added: Note that one can always find a normal finite index subgroup $\Gamma'\leq \Gamma$. Since $\mathcal{E}_{\Gamma}=(\mathcal{E}_{\Gamma'})^{\Gamma/\Gamma'}$ we readily see that if $\mathcal{E}_{\Gamma'}$ is affine then automatically $\mathcal{E}_{\Gamma}$ is affine being the quotient of affine variety by a finite group.

Answer by Donu Arapura for On the algberaicity of the universal elliptic curve associated to a torsion free subgroup

2012-05-29T17:43:10Z

The point is that by your construction $f:\mathcal{E}_\Gamma\to Y_\Gamma$ is an elliptic surface with a section $\sigma$. So as Keerthi said in the comments, $f$ is projective. In fact the relative divisor $3\sigma$ gives the Weirstrass embedding $\mathcal{E}_\Gamma\to \mathbb{P}(f_*\mathcal{O}(3\sigma))$ ). (Ben's comment shows that projectivity can fail without a section.) From here, quasiprojectivity is straight forward: If $V$ is an extension of $f_*\mathcal{O}(3\sigma)$ to a vector bundle on the smooth projective closure of $ Y_\Gamma$, then the closure of $\mathcal{E}_\Gamma$ in $\mathbb{P}(V)$ is a projective variety. Therefore $\mathcal{E}_\Gamma$ is quasiprojective.

Given the endless stream of comments, perhaps I should add a few words of clarification:

Since $Y=Y_\Gamma$ is noncompact, any vector bundle such as $f_*\mathcal{O}(3\sigma)$ is in fact (analytically) trivial. This is definitely overkill, but you can use Grauert, "Analytichse Faserungen..." Math. Ann 1958.
Thus $\mathbb{P}(\mathcal{O}(3\sigma))\cong Y\times \mathbb{P}^2$. This embeds into $\bar{Y}\times \mathbb{P}^2$, where $\bar Y$ is a (the) smooth projective compactification of $Y$.
We can take the closure of $\mathcal{E}_\Gamma$ to get a projective variety, and the quasiprojectivity of this family follows easily.
Note that the fibres of the closure $\overline{\mathcal{E}_\Gamma}$ may be singular.
If the Hopf surface had a section, it would lift to a rational curve in $\mathbb{C}^2-\lbrace 0\rbrace$. Well, I'll let you think about why that might be a problem.

OK, I guess there some issues.... I'm converting this to CW. Anyone, who wants to fix this is welcome to. I've got to finishing my refereeing....

Answer by Misha for On the algberaicity of the universal elliptic curve associated to a torsion free subgroup

2012-05-29T18:22:52Z

Here is an alternative take on Donu's argument: Removing the image $\sigma$ of a section of $E_\Gamma$ allows one to regard a fiber of $E_{\Gamma}$ as a once-punctured torus $S$. (In order to construct a section, use the standard upper half-space model of the Teichmuller space of the tori, so that each marked torus is identified with the fundamental parallelogram $P$ with vertices $0, 1, \omega, \omega+1\in {\mathbb C}$. Then the section is given by the map $\omega\to 0\in P$.) Then $M=E_{\Gamma}\setminus \sigma$ is isomorphic to the quotient of the Teichmuller space $T(S)$ of $S$ by a finite-index torsion-free subgroup $Mod^o_S$ of the mapping class group $Mod_S$ of $S$. Now, this is a general fact (Deligne-Mumford, et al) that $T(S)/Mod^o_S$ is quasi-projective (for any Riemann surface of finite type). It is not hard to see that DM compactification of $T(S)/Mod^o_S$ is our case will add (among other things) the curve $\sigma$ back to $M$, thus, providing a projective compactification of $M$.

In the special case you are interested in, it seems that quasi-projectivity of $E_{\Gamma}$ was first proven by Kodaira (On compact analytic surfaces. II, III), at least, Shioda (On elliptic modular surfaces, 1972) attributes the result to him.

Addendum: As an alternative to this argument, one can use Ron Livne's thesis "On certain covers of the universal elliptic curve". Livne proves that for every level $N\ge 5$ congruence subgroup $\Gamma(N)$ in the modular group $SL(2, {\mathbb Z})$, the universal elliptic curve over $E(N):={\mathbb H}^2/\Gamma(N)$ admits a degree $d\ge 2$ cyclic branched cover $E_d(N)$, so that the latter admits a compactification $X_d(N)$ (compatible with branched cover), so that $X_d(N)$ is a general type projective surface. Thus, every $E_\Gamma$ admits a finite regular cover which is biholomorphic to a finite cover over one of the $E_d(N)$'s. Thus, $E_\Gamma$ is quasi-projective.

Livne also refers to Mumford's paper "Prym varieties. I." Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 325–350. Academic Press, New York, 1974, for a direct proof of quasi-projectivity of $E(N)$'s. I do not have access to Mumford's paper, so I cannot say for sure.