N. Vladov R-P. Holzapfel with Appendices by A. Piñeiro
Picard-Einstein Metrics and Class Fields Connected with Apollonius Cycle
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

11G15 Complex multiplication and moduli of abelian varieties, See also {14K22}

11G18 Arithmetic aspects of modular and Shimura varieties, See also {14G35}

ZDM: 11H56
PACS: 14D05
CR: 11R11
Abstract: We define Picard-Einstein metrics on complex algebraic surfaces as
Kähler-Einstein metrics with negative constant sectional
curvature pushed down from
the unit ball via Picard modular groups allowing degenerations along cycles.
We demonstrate how the tool of orbital heights, especially the
Proportionality
Theorem presented in [H98], works for detecting such orbital cycles on the
projective plane. The simplest cycle we found on this way is supported by a
quadric and three tangent lines (Apollonius configuration). We give a
complete
proof for the fact that it belongs to the congruence subgroup of level 1+i
of the full Picard modular group of Gauß numbers together with precise
octahedral-
symmetric interpretation as moduli space of an explicit Shimura family of
curves of genus 3. Proofs are based only on the Proportionality Theorem and
classification results for hermitian lattices and algebraic surfaces.

Keywords: algebraic curves, moduli space, Shimura surface, Picard modular group, arithmetic group, Gauß lattice, Kähler-Einstein metric, negative constant curvature, unit ball