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Distribution of Kaneko's val function
Abstract
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)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.
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BibTeX
@misc{matsusaka2026distribution,
title = {{Distribution of Kaneko's val function}},
author = {Toshiki Matsusaka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00560},
eprint = {2609.00560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00560},
}
RIS
TY - UNPB
TI - Distribution of Kaneko's val function
AU - Toshiki Matsusaka
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00560
UR - https://arxiv.org/abs/2609.00560
AB - 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)_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.
ER -