Isomorphic Lattices, complex Tori and their relation to Jacobians Let $g >1$ a natural number and $\mathbb{C}^g$ complex vector space which
is isomorphic to $\mathbb{R}^{2g}$ is real vector space.
An additive subgroup $\Gamma \subset \mathbb{C}^g$ is called a
lattice if there exist $2g$ vectors $\gamma_1,... \gamma_{2g}$,
which are linearly independent over $\mathbb{R}$ such that
$\Gamma= \mathbb{Z} \gamma_1 + ... + \mathbb{Z} \gamma_{2g}$.
Let $\Gamma, \Gamma' \subset \mathbb{C}^g$ be two lattices with
$\Gamma= \mathbb{Z} \gamma_1 + ... + \mathbb{Z} \gamma_{2g}$ and
$\Gamma'= \mathbb{Z} \gamma' _1 + ... + \mathbb{Z} \gamma' _{2g}$. Is there a characterization when two quotient
groups $\mathbb{C}^g / \Gamma$ and $\mathbb{C}^g / \Gamma'$
are isomorphic as abelian groups in dependence of a certain relation between lattices $\Gamma$ and $\Gamma'$?
My first guess was  $\mathbb{C}^g / \Gamma \cong \mathbb{C}^g / \Gamma'$
if and only if there exist a $M \in GL_{2g}(\mathbb{Z})$ with
$M \cdot \Gamma = \Gamma' $ and $M \cdot \gamma_i = \gamma_i '$. Or should I require that moreover $M$ lives in $O_{2g}(\mathbb{Z})$,  $O_{2g}(\mathbb{Z})$ or even a scalar matrix $c \cdot Id$ with $c \in \mathbb{C} \backslash \{0\}$?
My motivation is  my question about Jacobians of Riemann surfaces
from Forster's Lectures on Riemann Surfaces. We have a
compact Riemann surface $X$ of genus $g$ and Forster's construction
of the Jacobian $Jac(X)$ bases on an explicit choice of basis
$\omega_1,..., \omega_g$ of the $\mathbb{C}$-space of holomorphic
$1$-forms $\Omega (X)$. Forster shows that the subspace of $\mathbb{C}^g$
consisting of all vectors
$$(\int_{\alpha} \omega_1, \int_{\alpha} \omega_2, ...
\int_{\alpha} \omega_g)$$
where $α$ runs through the fundamental group $\pi(X)$ form a lattice
$\Gamma= \mathbb{Z} \gamma_1 + ... + \mathbb{Z} \gamma_{2g}
\operatorname{Per}(\omega_1,..., \omega_g) \subset \mathbb{C}^{g}$ and
Jacobian is defined by $Jac(X):= 
\mathbb{C}^g/ \operatorname{Per}(\omega_1,..., \omega_g)$. At first glace
this definition seems to be bad because of a
choice of the basis $\omega_1,..., \omega_g$. But Forster remarked also
without providing a proof that a choice of a different basis leads to an isomorphic
$Jac(X)$.
That is I have to know firstly when two quotients
$\mathbb{C}^g / \Gamma$ and $\mathbb{C}^g / \Gamma'$ with lattices
$\Gamma$ and $\Gamma'$ are considered as isomorphic abelian groups
(I assume that Forster not consider they additionally as compact complex manifolds, or what type of isomorphy Forster consider) and why choosing different basis'
gives isomorphic Jacobians?
 A: One way to describe the 1-dimensional result is to say that if $f: X\to X'$ is a biholomorphic map of two elliptic curves $X={\mathbb C}/\Gamma, X'=  {\mathbb C}/\Gamma'$, then:
Each lift $F$ of $f$ to ${\mathbb C}$ is an invertible complex-affine map $z\mapsto az+b$, equivariant with respect to an isomorphism of free abelian groups $\phi: \Gamma\to \Gamma'$, i.e.:
$$
F\circ \gamma= \phi(\gamma)\circ F, \forall \gamma\in \Gamma. 
$$
Conversely, each affine map $F$ as above descends to a biholomorphic map $f: X\to X'$.
Exactly the same works in higher dimensions when $\Gamma, \Gamma'$ are lattices in ${\mathbb C}^n$, except, of course, invertible complex-affine  maps are given by
$$
z\mapsto Az + b, A\in GL(n, {\mathbb C}), b\in {\mathbb C}^n. 
$$
A proof is rather straightforward: Lift $f: X\to X'$ to a biholomorphic map
$$
F: {\mathbb C}^n\to {\mathbb C}^n
$$
which is then equivariant with respect to an isomorphism $\phi: \Gamma\to \Gamma'$,
$$
F\circ \gamma \circ F^{-1}= \phi(\gamma), \forall \gamma\in \Gamma. 
$$
Differentiating the equivariance condition
$$
F\circ \gamma= \phi(\gamma)\circ F, \forall \gamma\in \Gamma, 
$$
using the Chain Rule we obtain that
$$
DF \circ \gamma= DF, \forall \gamma\in \Gamma.
$$
Hence, the Jacobian derivative $DF$ is a $\Gamma$-invariant holomorphic mapping
${\mathbb C}^n\to {\mathbb C}^n$, hence, descends to a holomorphic map $X\to {\mathbb C}^n$, which then has to be constant by the compactness of $X$. Thus, the map $z\mapsto DF(z)$ is also constant. In other words, $F$ is a complex-affine map. qed
Traditionally, the equivariance condition is written in its matrix form, by choosing a free basis of $\Gamma$ and specifying its image in $\Gamma'$ under $\phi$. Feel free to rewrite the above equations the same way, personally, I do not like this since it depends on auxiliary choices.
