Intermediate Jacobian and Some Arithmetic Properties of Kummer-surface-type CY 3-folds

In this article, we examine the arithmetic aspect of the Kummer-surface-type CY 3-folds $\widehat{T\mathrm{/}G}$, characterized by the crepant resolution of 3-torus-orbifold $T\mathrm{/}G$ with only isolated singularities. Up to isomorphisms, there are only two such space $\widehat{T\mathrm{/}G}$ with $\mathrm{\mid }G\mathrm{\mid }=3,7$, and both $T$ carrying the structure of triple-product structure of a CM elliptic curve. The (Griffiths) intermediate Jacobians of these $\widehat{T\mathrm{/}G}$ are identified explicitly as the corresponding elliptic curve appeared in the structure of $T$. We further provide the $\text{QZ}$-structure of $\widehat{T\mathrm{/}G}$ and verify their modularity property.