Extremal Properties of Families of Lattices

Abstract

In this chapter (except in the last section) we consider families ℱ of lattices which are orbits of one of them under the action of a subgroup 𝓖 of GL (E), which we assume to be closed, invariant under transposition, such that the connected component of its unit element has finite index, and which moreover satisfies the following condition: either 𝓖 has determinant 1, or 𝓖 contains all positive homothetic transformations. We study ℱ-extreme lattices, i.e. lattices in ℱ on which the Hermite invariant attains a local maximum among lattices which belong to ℱ. Thanks to our hypotheses on 𝓖, we can use its structure as a Lie group, and more precisely the existence of a tangent space at the origin and of an exponential map.