The isotropic lines of Z_{d}^{2}

TL;DR

This paper characterizes isotropic lines in the lattice Z_{d}^{2} as Lagrangian submodules, describes their structure and orbits under SL(2,Z_d), and explores related group actions.

Contribution

It provides a detailed classification of isotropic lines in Z_{d}^{2}, including their enumeration, orbit structure, and associated group actions, advancing understanding of lattice symmetries.

Findings

Isotropic lines are Lagrangian submodules of Z_{d}^{2}

Number and orbit structure of isotropic lines are explicitly described

Group actions on related groups are developed

Abstract

We show that the isotropic lines in the lattice Z_{d}^{2} are the Lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice. The set of isotropic lines decompose into orbits under the action of SL(2,Z_d). We give an explicit description of those orbits as well as their number and their respective cardinalities. We also develop two group actions on the group \Sigma_{D}(M) related to the topic.