Spherical Designs and Heights of Euclidean Lattices

TL;DR

This paper explores the relationship between spherical designs and the extremal properties of lattice height functions, identifying conditions under which lattices are stationary points and providing computational results up to dimension 7.

Contribution

It establishes that lattices with all layers forming spherical 2-designs are stationary points of the height function and introduces a modular forms approach to identify such lattices.

Findings

Lattices with spherical 2-design layers are stationary points of the height function.

A modular forms strategy helps identify lattices with spherical design properties.

Explicit computations for dimensions up to 7 are provided.

Abstract

We study the connection between the theory of spherical designs and the question of extrema of the height function of lattices. More precisely, we show that a full-rank n-dimensional Euclidean lattice, all layers of which hold a spherical 2-design, realises a stationary point for the height function, which is defined as the first derivative at 0 of the spectral zeta function of the associated flat torus. Moreover, in order to find out the lattices for which this 2-design property holds, a strategy is described which makes use of theta functions with spherical coefficients, viewed as elements of some space of modular forms. Explicit computations in dimension up to 7, performed with Pari/GP and Magma, are reported.