To achieve this, we first consider Problem \[complex:pic\] and then use the above mentioned work of Shimura and Matsushima to relate Problems \[complex:pic\] and \[cp-prod-intro-prob\] *(b)*. As it turns out, Theorem \[existence:prop\] is a special case of Theorem \[Pair:index:theorem\]. We include the proof of Theorem \[existence:prop\], even though it is similar to that of Theorem \[Pair:index:theorem\], because it is elementary and solidifies the main ideas used in the proof of the more general result.

### Real multiplication and line bundles {#PIC:number:fields}

Let $K$ denote a totally real number field of degree $g$ over $\QQ$, let $\sigma_1,\dots, \sigma_g$ denote the embeddings of $K$ into $\RR$, and let ${{\mathcal O}}_K$ denote the ring of integers of $K$.

A complex torus $X$ *admits multiplication by $K$* if there exists an embedding $$K \hookrightarrow \operatorname{End}(X)\otimes_\ZZ \QQ \text{.}$$ Let $\mathfrak{h}$ denote the upper half plane, let ${\mathbf{z}}=(z_1,\dots, z_g)\in \mathfrak{h}^g$, and let $$\Lambda_{\mathbf{z}}:= \{(z_1 \sigma_1(\alpha) + \sigma_1(\beta),\dots, z_g \sigma_g(\alpha)+\sigma_g(\beta)) : \alpha, \beta \in {{\mathcal O}}_K \} \subseteq \CC^g \text{.}$$ Then $\Lambda_{\mathbf{z}}$ is the $\ZZ$-linear span of an $\RR$-basis for $\CC^g$ and the quotient $X_{\mathbf{z}}:= \CC^g / \Lambda_{\mathbf{z}}$ is a complex torus which has multiplication by $K$, [@BL §9.2].

\[existence:prop\] Let $p$ and $q$ be nonnegative integers 