On modular ball-quotient surfaces of Kodaira dimension one

2010/09/28 by Momot, Aleksander
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1009.5620

Abstract

Let Γ⊂ PU(2,1) be a lattice which is not co-compact, of finite Bergman-covolume and acting freely on the open unit ball B ⊂ ℂ². Then the compactification X = Γ∖ B is a smooth projective surface with an elliptic compactification divisor D = X ∖ (Γ∖ B). In this short note we discover a new class of unramified ball-quotients X. We consider ball-quotients X with kod(X) = h¹(X, 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 ∈ E[6] such that \Z x+\Z y = E[6].

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