On Modular Ball-Quotient Surfaces of Kodaira Dimension One

Abstract EN

Let Γ⊂PU(2,1) be a lattice which is not co-ompact, of finite covolume with respect to the Bergman metric and acting freely on the open unit ball B⊂ℂ2.

Then the toroidal compactification X=Γ\B¯ is a projective smooth surface with elliptic compactification divisor D=X\(Γ\B).

In this short note we discover a new class of unramifed ball quotients X.

We consider ball quotients X with kod(X)=1 and h1(X,OX)=1.

We prove that each minimal surface with finite Mordell-Weil group in the class described admits an étale covering which is a pull-back of X6(6).

Here X6(6) denotes the elliptic modular surface parametrizing elliptic curves E with 6-torsion points x,y which generate E[6].