Geodesic
Complex Hyperbolic Surfaces of Abelian Type
Serdica Mathematical Journal, Tome 30 (2004) no. 2-3, pp. 207-238

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We call a complex (quasiprojective) surface of hyperbolic type, iff – after removing finitely many points and/or curves – the universal cover is the complex two-dimensional unit ball. We characterize abelian surfaces which have a birational transform of hyperbolic type by the existence of a reduced divisor with only elliptic curve components and maximal singularity rate (equal to 4). We discover a Picard modular surface of Gauß numbers of bielliptic type connected with the rational cuboid problem. This paper is also necessary to understand new constructions of Picard modular forms of 3-divisible weights by special abelian theta functions.
Keywords: Algebraic Curve, Elliptic Curve, Algebraic Surface, Shimura Variety, Arithmetic Group, Picard Modular Group, Gauß Numbers, Congruence Numbers, Negative Constant Curvature, Unit Ball, Kähler-Einstein Metrics 
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Holzapfel, R. Complex Hyperbolic Surfaces of Abelian Type. Serdica Mathematical Journal, Tome 30 (2004) no. 2-3, pp. 207-238. http://geodesic.mathdoc.fr/item/SMJ2_2004_30_2-3_a6/