On modular ball-quotient surfaces of Kodaira dimension one

Let $\Gamma \subset \mathbf{PU}(2,1)$ be a lattice which is not co-compact, of finite Bergman-covolume and acting freely on the open unit ball $\mathbf{B} \subset \mathbb{C}^2$. Then the compactification $X = \bar{\Gamma \setminus \mathbf{B}}$ is a smooth projective surface with an elliptic compactification divisor $D = X \setminus (\Gamma \setminus \mathbf{B})$. In this short note we discover a new class of unramified ball-quotients $X$. We consider ball-quotients $X$ with $kod(X) = h^1(X, \mathcal{O}_X) = 1$. We prove that all minimal surfaces with finite Mordell-Weil group in the class described become after an etale base change pull-backs of the elliptic modular surface which parametrizes triples $(E,x,y)$ of elliptic curves $E$ with 6-torsion points $x,y \in E[6]$ such that $\Z x+\Z y = E[6]$.