and $\chi(L \otimes M)$ are nonzero and ${\operatorname{i}}(L \otimes M) = {\operatorname{i}}(L) + {\operatorname{i}}(M)$.}$$

In this paper we study questions surrounding cup-product maps of the form $$\label{intro:PIC:cup-product}
{\mathrm{H}}^{{\operatorname{i}}(L)}(X,L) \otimes {\mathrm{H}}^{{\operatorname{i}}(M)}(X,M) \xrightarrow{\cup} {\mathrm{H}}^{{\operatorname{i}}(L \otimes M)}(X, L \otimes M)$$ where $L$ and $M$ are line bundles on $X$ for which condition holds — condition implies that the source and target space of the map are nonzero.

\[cp-prod-intro-prob\] Let $X$ be an abelian variety of dimension $g$.

1.  [Let $L$ and $M$ be line bundles on $X$ which satisfy condition . Describe the image of the cup-product map . For instance, is it nonzero?]{}

2.  [For what $p,q \in \NN$, $p + q {\leqslant}g$, does $X$ admit line bundles $L$ and $M$ such that $\chi(L)$, $\chi(M)$, and $\chi(L \otimes M)$ are nonzero, ${\operatorname{i}}(L) = p$, ${\operatorname{i}}(M) = q$, and ${\operatorname{i}}(L \otimes M) = p + q$?]{}

Over the complex numbers, a question related to Problem \[cp-prod-intro-prob\] *(b)* is

\[complex:pic\] Fix a lattice $\Lambda \subseteq \CC^g$ and $p, q \in \NN$. Do there exists Hermitian forms $$\text{$H_i : \CC^g \times \CC^g \rightarrow \CC$, for $i = 1,2$,}$$ whose imaginary parts are integral on $\Lambda$, and have the properties that

-   [the Hermitian forms $H_1$, 