TITLE: Is SL(2,C)/SL(2,Z) a quasi-projective variety?
QUESTION [22 upvotes]: Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).

Is $SL(2,\mathbb C)/SL(2,\mathbb Z)$ a quasi-projective variety?

The natural generalization of this question seems to be the following.  Let $G$ be a semisimple linear algebraic group over $\mathbb Q$.  Then $G(\mathbb Z)$ is well-defined up to taking a finite-index subgroup.  Thus we can ask the same question: is the complex manifold $G(\mathbb C)/G(\mathbb Z)$ a quasi-projective variety?

REPLY [26 votes]: No, the quotient is not quasi-projective. In the paper Invariant meromorphic functions on complex semisimple Lie groups by D. N. Ahiezer you can find the following result.

Theorem. Let $G$ be a connected semisimple linear algebraic group
  defined over the rationals and
  $\Gamma$ be a  subgroup of $G(\mathbb
> Q)$ which is Zariski dense in $G$.
  Then there are no invariant analytic
  hypersurfaces in $G$ invariant by the
  action of $\Gamma$. In particular, if
  the quotient $G/ \Gamma$ exists as a
  complex variety then every 
  meromorphic function on it is constant.

This implies that the quotient is not quasi-projective and even more: it cannot be holomorphically  embedded in any algebraic variety.