# Complex structures on torus

I am trying to understand the space of isomorphisms of lattices in $\mathbb R^2$. The set of bases is $\mathrm{GL}_2\mathbb R$ and now I would like to quotient out automorphisms of the lattices, which, if I believe are generated by the matrices $$\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \quad\text{and}\quad \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$$ corresponding to switching the basis vectors and having them point in the other direction. Denote the group generated by these two matrices $\mathrm{Aut}\,\Lambda$.

What space is $\mathrm{GL}_2\mathbb R/\mathrm{Aut}\,\Lambda$?

(Considering $\mathbb C=\mathbb R+i\mathbb R$ this is the moduli space of complex structures on the torus...?)

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you want rather $GL_2(\mathbb{R})/GL_2(\mathbb{Z})$ – user8268 Jul 24 '12 at 12:31
I want...? Could you explain what you mean? – Earthling Jul 24 '12 at 12:38
maybe you don't :) what I meant: two elements of $GL_2(\mathbb{R})$ (i.e. two bases) give the same lattice iff they give the same element of $GL_2(\mathbb{R})/GL_2(\mathbb{Z})$ – user8268 Jul 24 '12 at 12:41
How about $\begin{pmatrix}2&0\\0&2\end{pmatrix}$? Multiplying by this seems to give a bigger lattice... Maybe $SL_2(\mathbb Z)$ is really what I want, what do you think? – Earthling Jul 24 '12 at 12:44
that matrix is not in $GL_2(\mathbb{Z})$, as its inverse doesn't have integer coefficients. A matrix with integer coefficients is in $GL_2(\mathbb{Z})$ iff its determinant is $\pm1$. – user8268 Jul 24 '12 at 12:48
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