发表机构
Faculty of Mathematics, Kyushu University（九州大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明金子（Kaneko）的val函数在本原双曲共轭类按测地线长度排序时集中于720，对权0弱全纯模函数也有类似结论，结合等分布定理等完成证明。
AI 中文摘要
金子（Kaneko）的val函数定义为模曲面闭合测地线上椭圆模j函数的正则化周期积分。我们证明，当本原双曲共轭类按测地线长度排序时，其值集中在单点720。更一般地，对每个权为0的弱全纯模函数f，存在类似的集中结果，集中点由Atkin内积(f,1)_{At}给出。证明结合了Pollicott的等分布定理、连分数悬挂流的遍历性，并使用Bengoechea-Imamoglu的分解公式构造有界连续可观测量。
英文摘要
Kaneko's val function is defined as the normalized cycle integral of the elliptic modular $j$-function along closed geodesics on the modular surface. We prove that, when primitive hyperbolic conjugacy classes are ordered by geodesic length, its values concentrate at the single point 720. More generally, an analogous concentration result holds for every weakly holomorphic modular function $f$ of weight 0, with the concentration point given by Atkin's inner product $(f, 1)_{\mathrm{At}}$. The proof combines an equidistribution theorem following Pollicott with the ergodicity of a continued-fraction suspension flow and uses the decomposition formula of Bengoechea-Imamoglu to construct a bounded continuous observable.
Comments18 pages
