4 added 15 characters in body

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

Indeed let us write $X_i=\mathbb{C}^{g_i}/\Gamma_i$, X_i=V_i/\Gamma_i$, where$\Gamma_i$is a lattice . in the complex vector space$V_i$. Then$X_1 \times X_2 \cong V_1 \mathbb{C}^{g_1+g_2}/ times V_2/ \Gamma_1 \times \Gamma_2$and, by standard representation theory of abelian groups, the character group of$\Gamma_1 \times \Gamma_2$coincides with the direct product of the character groups of$\Gamma_1$and$\Gamma_2$. 3 added 4 characters in body Given two complex tori$X_1$and$X_2$, there is always a canonical isomorphism$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$, see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43. In fact, Indeed let us write$X_i=\mathbb{C}^{g_i}/\Gamma_i$, where$\Gamma_i$is a lattice. Then$X_1 \times X_2 \cong \mathbb{C}^{g_1+g_2}/ \Gamma_1 \times \Gamma_2$,Gamma_2$

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.

Given two complex tori $X_1$ and $X_2$, there is always a canonical isomorphism

$\widehat{X_1 \times X_2} \cong \widehat{X}_1 \times \widehat{X}_2$,

see for instance Birkenhake-Lange's book Complex Abelian Varieties, Exercise 11 page 43.

In fact, write $X_i=\mathbb{C}^{g_i}/\Gamma_i$, where $\Gamma_i$ is a lattice. Then

$X_1 \times X_2 \cong \mathbb{C}^{g_1+g_2}/ \Gamma_1 \times \Gamma_2$,

and, by standard representation theory of abelian groups, the character group of $\Gamma_1 \times \Gamma_2$ coincides with the direct product of the character groups of $\Gamma_1$ and $\Gamma_2$.

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