Table of Contents
ISRN Geometry
Volume 2011, Article ID 214853, 5 pages
http://dx.doi.org/10.5402/2011/214853
Research Article

On Modular Ball-Quotient Surfaces of Kodaira Dimension One

Departement Mathematik, ETH Zürich, HG J65, Rämistraße 101, 8092 Zürich, Switzerland

Received 12 April 2011; Accepted 30 April 2011

Academic Editors: D. Franco and E. H. Saidi

Copyright © 2011 Aleksander Momot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Ξ“ βŠ‚ 𝐏 𝐔 ( 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 βŠ‚ β„‚ 2 . Then the toroidal compactification 𝑋 = Ξ“ \ 𝐁 is a projective smooth surface with elliptic compactification divisor 𝐷 = 𝑋 \ ( Ξ“ \ 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 β„Ž 1 ( 𝑋 , π’ͺ 𝑋 ) = 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 𝑋 6 ( 6 ) . Here 𝑋 6 ( 6 ) denotes the elliptic modular surface parametrizing elliptic curves 𝐸 with 6-torsion points π‘₯ , 𝑦 which generate 𝐸 [6].