Tempered perfect lattices in the binary case

Abstract

A new algorithm for computing Hecke operators for SLn 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 SL2(Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.