Zeta functions of quadratic lattices of a hyperbolic plane

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.