Towards a Classification of Rigid Product Quotient Varieties of Kodaira Dimension 0

In this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group $G$. It is shown that only for $G = operatorname{He(3)}, mathbb Z_3^2$, and only for dimension $geq 4$ such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension $4$ is given. For the other finite groups a strong structure theorem for rigid quotients is proven.