Similar sublattices of planar lattices

TL;DR

This paper classifies similar sublattices of planar lattices using their multiplier rings, deriving generating functions and relating them to zeta functions of orders in imaginary quadratic fields.

Contribution

It provides a classification framework for similar sublattices via multiplier rings and derives Dirichlet series generating functions linked to zeta functions.

Findings

Classification of sublattices via multiplier rings

Derivation of Dirichlet series generating functions

Connection to zeta functions of imaginary quadratic orders

Abstract

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to various zeta functions of orders in imaginary quadratic fields.