Zeta functions of quadratic lattices of a hyperbolic plane

TL;DR

This work extends the zeta-function framework for counting sublattices from positive definite to indefinite binary lattices realized on a hyperbolic plane. By reducing to sublattices of the hyperbolic plane and exploiting inclusion-exclusion, Möbius inversion, and detailed divisor-structure analysis, the authors derive explicit formulas for the SL-zeta function $\zeta_{L}^{\mathrm{SL}}(s)$, including a convergent expression in terms of $B=[\mathbb{H}:L](\mathfrak{n}L)^{-1}$ and a refined combinatorial description involving $U_b^{2}$ and $c_b(\mathbf{w})$. The main contributions are (i) a closed form $\zeta_{L}^{\mathrm{SL}}(s)=\frac{\zeta(s-1)}{\zeta(s)}\sum_{b\mid B}(B/b)^{-s}\sum_{m=1}^{\infty}\frac{\#\{d^{2}\ (\bmod b): d\mid m,\gcd(d,b)=1\}}{m^{s}}$, together with an alternative $U_b^{2}$-based expression, (ii) the demonstration that the abscissa of convergence is $\Re(s)=2$ with a simple pole at $s=2$, and (iii) a framework showing that a large portion of proper equivalence classes are one-lattice. The latter is supported by explicit computations and asymptotic analysis of lattice counts. This advances a systematic understanding of zeta functions of quadratic lattices in the indefinite (hyperbolic) setting and provides practical formulas for exact evaluation and asymptotics.

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.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• proof
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof