this a pretty idiot question and of course there is a mistake in my way of thinking.
Let $E$ be a elliptic curve, $E (\mathbb{C}) \cong \mathbb{C} / \Lambda$, where $\Lambda = \langle \omega_1, \omega_2 \rangle \cong \mathbb{Z}^2$. Clearly, two elliptic curves $\mathbb{C} / \mathbb{\Lambda}_1$ and $\mathbb{C} / \mathbb{\Lambda}_2$ are isomorphic iff there exists a non-zero complex number $\alpha$ such that $\alpha\Lambda_1 = \Lambda_2$.
Clearly, there is an identification between lattices in $\mathbb{C}$ and $\text{Hom}^{+} (\mathbb{R}^2, \mathbb{C})$ (the set of $\mathbb{R}$-vector spaces morphisms such that the canonical basis of $\mathbb{R}^2$ has positive orientation under the image of the morphism). By the above remark, $\mathbb{C}^{*} \setminus \text{Hom}^{+} (\mathbb{R}^2, \mathbb{C}) \cong \mathfrak{h}$ would parametrize all isomorphic classes of lattices. However it's known to not be true, because there still need an action by $\text{SL}_2(\mathbb{Z})$.
What's wrong here?
Thanks in advance.
An element of your $\hom$ is not a lattice but «a lattice with a fixed basis». $SL_2$ is needed to identify such things which only differ in the bases.