Billiards, heights, and the arithmetic of non-arithmetic groups

• 成果类型: Article
• 署名作者: McMullen, Curtis T.
• 署名单位: Harvard University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01101-4
• 发表日期: 2022
• 页码: 1309-1351
• 关键词: teichmuller curves; triangle groups; geodesics; series

摘要

In this paper we introduce a new height on P-1 (K) associated to an Abelian variety with real multiplication by K, and use it to study non-arithmetic triangle groups, Teichintiller curves, and billiards in lattice polygons. Complementary results on matrix coefficients and measures are obtained using modular symbols. In particular, we show the matrix entries m of the classical Hecke group Delta(2, 5, infinity) are constrained by the condition that -gamma(-2)(m'/m) lies in a countable, closed semigroup S subset of [-1, 1] homeomorphic to omega(omega) + 1.