whose sum is less than or equal to $g$. Let ${\mathbf{z}}=(z_1,\dots, z_g)\in \mathfrak{h}^g$. The complex torus $X_{\mathbf{z}}$ admits line bundles $L$ and $M$ such that $L$, $M$, and $L\otimes M$ are non-degenerate, ${\operatorname{i}}(L) = p$, ${\operatorname{i}}(M) = q$, and ${\operatorname{i}}(L \otimes M) = p + q$.

First, note that every $\eta \in {{\mathcal O}}_K$ determines a Hermitian form $$H_{\eta}:\CC^g\times \CC^g\rightarrow \CC \text{ defined by } 
 H_{\eta}({\mathbf{x}},{\mathbf{y}}):= \sum\limits_{i=1}^{g}\frac{\sigma_i(\eta)}{\operatorname{Im} z_i} x_i \overline{y_i} \text{}$$ for ${\mathbf{x}}=(x_1,\dots, x_g)$ and ${\mathbf{y}}=(y_1,\dots, y_g) \in \CC^g$.

Also if ${\mathbf{x}}\text{, } {\mathbf{y}}\in \Lambda_{{\mathbf{z}}}$ and $$\text{${\mathbf{x}}= (z_1 \sigma_1(\alpha)+\sigma_1(\beta),\dots, z_g \sigma_g(\alpha)+\sigma_g(\beta))$,}$$ $$\text{${\mathbf{y}}= (z_1 \sigma_1(\gamma)+\sigma_1(\zeta),\dots, z_g \sigma_g(\gamma)+\sigma_g(\zeta))$,}$$ with $\alpha$, $\beta$, $\gamma$, and $\zeta \in {{\mathcal O}}_K$, then $$\operatorname{Im} H_{\eta}({\mathbf{x}},{\mathbf{y}}) = \operatorname{Tr}_{K/\QQ}(\eta \alpha \zeta) - \operatorname{Tr}_{K/\QQ}(\eta \beta \gamma) \text{.}$$ Thus, the imaginary part of $H_\eta$ is integral on $\Lambda_{\mathbf{z}}$.

Fix disjoint subsets $I,J \subseteq \{1,\dots, g\}$ of cardinalities $p$ and $q$ respectively. Let $U \subseteq \RR^{2g}$ be the subset consisting of those points $(x_1,\dots, x_g,y_1,\dots, y_g) \in \RR^{2g}$ such that $x_k < 0$ if $k \in I$, $x_k > 0$ if $k \not \in I$, $y_k < 0$ if $k \in J$, $y_k > 0$ if $k \not \in J$, and $x_k + y_k < 0$ for all $k \in I \cup 