Prime Geodesic Theorem for Arithmetic Compact Surfaces
Chenhao Tang, Han Wu, Jie Yang, Wenyan Yang
TL;DR
This work analyzes the prime geodesic theorem for arithmetic compact surfaces arising from division quaternion algebras, extending Koyama's $x^{7/10+\\epsilon}$-bound to principal congruence subgroups ${\\Gamma_{\\mathcal D}^1(\\mathfrak N)}$. It avoids the spectral Jacquet–Langlands machinery and instead builds a robust local–global framework of orbital-integral matching between ${\\rm GL}_2$ and its inner form via chain orders, translating adelic stable orbital integrals into congruence-subgroup counting data. The main achievement is an explicit global identity expressing the counting function $\\Psi_{\\Gamma_{\\mathcal D}^1(\\mathfrak N)}(x)$ as a linear combination of counting functions for Eichler-congruence subgroups on the matrix side, which, combined with known $x^{7/10+\\epsilon}$-type bounds there and a crucial product identity, yields the desired error term for the arithmetic lattice. This method connects non-Archimedean harmonic analysis, local embedding theory, and adelic trace techniques to obtain sharp, unconditional bounds with potential extensions to more general congruence subgroups and higher rank inner forms.
Abstract
We generalize Koyama's $7/10$ bound of the error term in the prime geodesic theorems to the principal congruence subgroups for quaternion algebras. Our method avoids the spectral side of the Jacquet--Langlands correspondences, and relates the counting function directly to those for the principal congruence subgroups of Eichler orders of level less than one.