Picard-Einstein Metrics and Class Fields Connected with Apollonius Cycle
Picard-Einstein Metrics and Class Fields Connected with Apollonius Cycle
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与阿波罗循环相关的皮卡德-爱因斯坦度量和类域
DOI：
10.18452/2681
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
N. Vladov
中科院分区：
文献类型：
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作者：
R. Holzapfel;A. Piñeiro;N. Vladov
我们将复杂代数曲面上的皮卡德-爱因斯坦度量定义为具有负恒定截面曲率的卡勒-爱因斯坦度量，该度量通过皮卡德模群从单位球向下推，允许沿周期退化。我们演示了轨道高度工具，特别是（H98）中提出的比例定理如何用于检测射影平面上的此类轨道周期。我们以这种方式发现的最简单的循环由二次曲线和三条切线（阿波罗配置）支持。我们给出了完整的证明，证明它属于高斯数的完整皮卡德模群的 1 + i 级同余子群，并具有精确的八面体对称解释，作为 3 格显式 Shimura 族曲线的模空间。证明仅基于比例定理以及厄米格和代数曲面的分类结果。 1
We define Picard-Einstein metrics on complex algebraic surfaces as Kahler-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 Gaus 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. 1