I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action.


Similarly, I have been told that in the higher dimensional cases, the symplectic group $Sp(2n, \mathbb{R})$ acts on some such space to give the moduli space of complex structures on higher dimensional complex tori. Is there a reference that covers this case in detail and gives explicit formulas for the action? 


In the 1-dimensional case, all complex tori can be realized as algebraic varieties, but this is not the case for higher dimensional complex tori. Does the action preserve complex structures that come from abelian varieties? 


Crossposted at <https://math.stackexchange.com/questions/345713/moduli-spaces-of-higher-dimensional-complex-tori> where it has been unanswered for awhile.



Moduli Spaces of Higher Dimensional Complex Tori
The fact that all $1$-dimensional tori are projective means care is sometimes needed in making analogies with higher dimensional tori. This is one of those times. The natural 'moduli space' of all $d$-dimensional complex tori constructed by Will is not very nice when $d>1$. For example, the action of $GL\_{2d}(\mathbb Z)$ on $GL\_d(\mathbb C)\backslash GL\_{2d}(\mathbb R)$ isn't properly discontinuous, so the resulting quotient space isn't Hausdorff (an observation due to Siegel).


A slightly better state of affairs is available if we begin by looking at the Siegel upper half-plane
$$ \mathcal H\_d = \{ \tau \in M\_d(\mathbb C) \colon \tau^t = \tau, \mathrm{Im}(\tau) > 0 \}. $$
The group $Sp(2d,\mathbb R)$ acts transitively on $\mathcal H\_d$ via
$$ \begin{pmatrix}A & B\\C & D\end{pmatrix} \tau = (A\tau+B)(C\tau+D)^{-1} $$
(this is well-defined) and the isotropy subgroup of $iI\_d$ is (essentially) $U(g)$. Thus we can view $\mathcal H\_d$ as $Sp(2d,\mathbb R)/U(d)$. Now a point $\tau \in \mathcal H\_d$ gives us a complex torus $\mathbb C^d/(\mathbb Z^d + \mathbb Z^d \tau)$ -- but this torus is not an arbitrary one: it comes with a "principal polarization". Moreover the natural action of $Sp(2d,\mathbb Z)$ on $\mathcal H\_d$ (which, by the way, *is* properly discontinuous) preserves the isomorphism class of the corresponding torus as well as its principal polarization. Thus the points of the space $\mathcal H\_d/Sp(2d,\mathbb Z)$ are in bijection with isomorphism classes of principally polarized $d$-dimensional complex tori (these are abelian varieties). Note that if $d=1$ we're reduced to the familiar modular curve $\mathcal H/SL(2,\mathbb Z)$ which parametrizes all $1$-dimensional tori.


For (much) more on this, see II.4 of Mumford's Tata Lectures on Theta I or van der Geer's chapter in The 1-2-3 of Modular Forms.


