Actually I have a few related questions.


Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}\_2(\mathbb{Z})\backslash\mathcal{H}$.


I know $Y(1)$ is only a coarse moduli space, so there isn't a universal elliptic curve over it, but **does there exist an elliptic surface over it such that the fiber above every point is the elliptic curve corresponding to that point?**


Now, I know that above the open set $Y(1) \setminus \{i, e^{2\pi i/3}\}$, there is such an elliptic surface. My second question is: **Does there exist an open cover of $Y(1)$ such that above each open set in the cover, there exists an elliptic surface with the above property?** (I'm not asking for the surfaces to glue.)


Complex analytically, one can consider the space:
$$\mathcal{H}\times\mathbb{C}$$
On this space we have an action (on the right) by $\mathbb{Z}\times\mathbb{Z}$, acting via $$(\tau,x)\cdot(a,b) = (\tau, x + a\tau + b),$$
and an action (on the left) by $\text{SL}\_2(\mathbb{Z})$, acting via
$$\gamma\cdot(\tau,x) = (\gamma\tau, x)$$
where $\gamma$ acts on $\mathcal{H}$ by fractional linear transformations. Since intuitively, $\mathbb{Z}\times\mathbb{Z}$ acts on $\mathcal{H}\times\mathbb{C}$ "discretely", so the quotient $\mathcal{H}\times\mathbb{C}/\mathbb{Z}\times\mathbb{Z}$ should be a complex manifold, and is essentially an elliptic surface over $\mathcal{H}$.


If $\Gamma \subset \text{SL}\_2(\mathbb{Z})$, then we may also try to form the quotient 
$$\mathbb{E}(\Gamma) := \Gamma\backslash\mathcal{H}\times\mathbb{C}/\mathbb{Z}\times\mathbb{Z}.$$
Intuitively, if $\Gamma$ has no elliptic elements, then it should act "discretely" on $\mathcal{H}\times\mathbb{C}/\mathbb{Z}\times\mathbb{Z}$ and the quotient ought to be a manifold. Hence, if $\Gamma = \Gamma(2)$ (ie, matrices congruent to the identity mod 2), then since $\Gamma(2)$ has no elliptic elements, shouldn't $\mathbb{E}(\Gamma(2))$ be a (complex) manifold? If it is, then it's a complex manifold above $Y(2) := \Gamma(2)\backslash\mathcal{H}$, which is again only a coarse moduli scheme, and hence has no universal elliptic curve. In this case, **my third question is: is $\mathbb{E}(\Gamma(2))$algebraic? and if it is, can someone describe heuristically how it's different from a universal elliptic curve over $Y(2)$, if one existed?**


The context for these questions comes from me trying to understand why Katz's definition of modular forms for $\Gamma(N)$ (in his paper on $p$-adic modular forms) is properly a generalization of the analytic definition of modular forms. In particular, I'm trying to understand why they must give holomorphic functions on $\mathcal{H}$. I see why this must be the case when $N \ge 3$, since then you have a universal elliptic curve, and in this case holomorphicity is a result of the required compatibility with base change, but in the case of $N = 1$ and $N = 2$, I'm still a little confused. Relevant references would be appreciated as well.


thanks,


* will



Is there an elliptic surface over $Y(1)$?
1 The last exercise of Hartshorne chapter 4 section 4 proves that there is no elliptic surface over $\mathbb A^1$ with nonconstant $j$ invariant. Obviously this is a special case.


2 No as well. Put the fiber in Weirstrass form near $j=0$. Then one can write it as $y^2=x^3-g\_2x-g\_3$, where $g\_2$ and $g\_3$ are both functions of $j$ that are well-defined at $j=0$. Furthermore clearly $g\_3$ is nonvanishing at $j=0$ and $g\_2$ is vanishing at $j=0$. Then computing the $j$ invariant near $j=0$ as $g\_2^3/(4g\_2^3-27g\_3^2)$ up to a constant, we see it vanishes to at least third order, an obvious falsehood.


3 Yes, you're thinking of the Legendre family $y^2=x(x-1)(x-\lambda)$. The Legendre family is universal up to a quadratic twist - every family with full level two structure is a pullback of Legendre, up to a quadratic twist. So it is as close an approximation to the universal elliptic curve over $Y(2)$ as you can get in the category of schemes.


