On modular ball-quotient surfaces of Kodaira dimension one

Let $\mathrm{\Gamma }\subset \mathbf{P}\mathbf{U}(2,1)$ 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$\mathrm{.}Thenthecompactification$X = \bar{\Gamma \setminus $\mathbf{B}$}$ is a smooth projective surface with an elliptic compactification divisor $D=X\setminus (\mathrm{\Gamma }\setminus \mathbf{B})$. 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 $\mathbb{Z}x+\mathbb{Z}y=E[6]$.