Zeta functions of quadratic lattices of a hyperbolic plane

2025/05/01 by Kim, Daejun, Lee, Seok Hyeong, Lee, Seungjai
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.00484

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.

Citations

• Zeta functions enumerating subforms of quadratic forms
• Isolations of the sum of two squares from its proper subforms
• Can we recover an integral quadratic form by representing all its subforms?
• Isolations of cubic lattices from their proper sublattices

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