The harmonic strength of shells of lattices and weighted theta series

TL;DR

This paper explores the harmonic strength of lattice shells, linking it to the non-vanishing of Fourier coefficients of elliptic cusp forms, with specific focus on the D4 lattice and related root lattices.

Contribution

It establishes a connection between the harmonic strength of lattice shells and the non-vanishing of Fourier coefficients, providing new insights for root lattices and lattice design theory.

Findings

Identifies the relationship between harmonic strength and Fourier coefficients.

Provides examples from root lattices.

Highlights the non-vanishing problem for elliptic cusp forms.

Abstract

This is the write-up of a talk given in RIMS conference ``Analytic and arithmetic aspects of automorphic representations", where I outlined two kinds of different results related to the D4 lattice, obtained in a joint work with Hirao and Nozaki. In general, from the design theoretical viewpoints, we are interested in determining the harmonic strength of finite subsets on the unit sphere. When considering the shells of a lattice, this problem shifts to the non-vanishing problem of the Fourier coefficients of elliptic cusp forms. Various instances, mostly from root lattices, are listed in this note.