# Counting problems for geodesics on arithmetic hyperbolic surfaces

## TL;DR

This paper explores the relationship between geodesic length spectra and commensurability of arithmetic hyperbolic surfaces, providing quantitative bounds on non-commensurable surfaces sharing spectral subsets.

## Contribution

It introduces new quantitative results on the maximum number of non-commensurable surfaces sharing a fixed spectral subset, advancing understanding of spectral geometry.

## Key findings

- Bound on the number of non-commensurable surfaces sharing spectral data
- Quantitative relationships between length spectra and commensurability
- Insights into spectral sharing phenomena among hyperbolic surfaces

## Abstract

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra all contain a fixed (finite) set of nonnegative real numbers.