Tempered perfect lattices in the binary case

Abstract

A new algorithm for computing Hecke operators for ${\mathrm{SL}}_n$ was introduced in [14]. The algorithm uses tempered perfect lattices, which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. 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 ${\mathrm{SL}}_2\left(Z\right)$ and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.