Extremality and designs in spaces of quadratic forms

TL;DR

This paper develops a general framework unifying various characterizations of extremal quadratic forms and lattices, extending Voronoi's and Venkov's criteria through a notion of design, and explores extremality relative to the Epstein zeta function.

Contribution

It introduces a broad, unified framework that generalizes existing extremality criteria for quadratic forms and lattices using design theory.

Findings

Unified framework for extremality in quadratic forms and lattices.

Extension of Voronoi and Venkov criteria via design notions.

Application to Epstein zeta function extremality.

Abstract

A well known theorem of Voronoi caracterizes extreme quadratic forms and Euclidean lattices, that is those which are local maxima for the Hermite function, as perfect and eutactic. This caracterization has been extended in various cases, such that family of lattices, sections of lattices, Humbert forms, etc. Moreover, there is a criterion for extreme lattices, discovered by Venkov, formulated in terms of spherical designs which has been extended in the case of Grassmannians and sections of lattices. In this article, we define a general frame, in which there is a ``Voronoi characterization'', and a ``Venkov criterion'' through an appropriate notion of design. This frame encompasses many interesting situations in which a ``Voronoi characterization'' has been proved. We also discuss the question of extremality relatively to the Epstein zeta function, and we extend to our frame a characterization of final zeta-extremality formulated by Delone and Ryshkov and a criterion in terms of designs found by Coulangeon.