Zeta functions of quadratic lattices of a hyperbolic plane

TL;DR

This paper investigates zeta functions associated with quadratic lattices in hyperbolic planes, deriving explicit formulas and revealing that many classes are represented by a single lattice.

Contribution

It provides explicit formulas and a combinatorial method for computing zeta functions of quadratic lattices in hyperbolic planes, highlighting their analytic properties.

Findings

Many proper classes are one-lattice classes.

Explicit formulas for zeta functions are derived.

A combinatorial approach to computation is developed.

Abstract

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We derive explicit formulas for the associated zeta functions and obtain a combinatorial way to compute them. Their analytic properties lead to the intriguing consequence that a large proportion of proper classes are one-lattice classes.