Tempered Perfect Lattices in the Binary Case

TL;DR

This paper classifies tempered perfect lattices in the plane, advancing the understanding of their structure and relation to Hecke operators for SL(2,Z) and its subgroups, with implications for number theory.

Contribution

It provides the first complete classification of tempered perfect lattices in two dimensions, linking lattice theory with class field theory of imaginary quadratic fields.

Findings

Complete classification of tempered perfect lattices in the plane.

Connection established between lattices and class field theory.

Implications for Hecke operators in SL(2,Z).

Abstract

A new algorithm for computing Hecke operators for SL(n,Z) was introduced by MacPherson, McConnell in 2020. The algorithm uses tempered perfect lattices, which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL(2,Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.