Linear independence of theta series of positive-definite, unimodular even lattices

TL;DR

This paper establishes bounds on the minimal degree for which theta series of certain lattices are linearly independent, contributing to the understanding of lattice invariants in number theory.

Contribution

It provides new bounds on the minimal degree ensuring linear independence of theta series for positive-definite, unimodular even lattices of rank at least 24.

Findings

Bounds between m/2 and 3m/4 for minimal degree g

Linear independence of theta series in specified lattice classes

Enhanced understanding of lattice theta series structure

Abstract

We show that the minimal $g$ for which the degree $g$ theta series of positive-definite, unimodular even lattices of rank $m\geq24$ are linearly independent is bounded between $\frac{m}{2}$ and $\frac{3m}{4}$.