@Misha, do you have a nice reference (which covers the case that I'm interested in) which shows in details why is $T(S)/Mod_S^{\circ}$ algebraic. I would like to see how one uses torsion-freeness in order to deduce algebraicity, which is the whole point in my question.

the 3rd $\mathbf{C}$ should be replaced by $\mathbb{P}^1(\mathbf{C})$ of course.

Donu, I think I'm missing something important here. Say that you take the universal elliptic curve $E\rightarrow \mathbf{C}-\{0,1728\}$, then you cannot extend your vector bundle over the smooth projective closure of $\mathbf{C}-\{0,1728\}$ which is $\mathbf{C}$, since there is no universal elliptic curve over $\mathbf{C}$. So may be you could develop a little bit what you mean when you say extending the vector bundle to the smooth projective locus.

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...

@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..

I see, so I'll try to sketch the argument following your comment

So just to let you know that I posted a continuation of your "quite generous" post on quadratic forms in characteristic 2! see mathoverflow.net/questions/98003/…

Dear Igor, this is indeed a nice document but I could not dig the answer to the question

So I replaced family of elliptic curves for family of curves!

You are right I was a bit careless in the set up

Thanks Keith, it is good to have it for future reference! So what should one do in characteristic $2$ in order to save the statement? The naive definition for the inner product fails since we divide by $2$, moreover one should also replace the isotropy group by something a bit larger but what would be such a replacement...