Automorphisms of extremal unimodular lattices in dimension 72

TL;DR

This paper identifies the unique extremal even unimodular lattice in dimension 72 with a large automorphism, using advanced computational methods to narrow down possible automorphisms.

Contribution

It demonstrates the uniqueness of the extremal lattice with a large automorphism in dimension 72 through extensive Magma computations.

Findings

The extremal lattice in dimension 72 admits a unique large automorphism.

Computational methods confirm the lattice's uniqueness among extremal lattices.

The study refines understanding of automorphism groups of high-dimensional lattices.

Abstract

The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in {\sc Magma} using the very sophisticated algorithm for computing class groups of algebraic number fields written by Steve Donnelly it is shown that the extremal even unimodular lattice $\Gamma _{72}$ from \cite{dim72} is the unique extremal even unimodular lattice of dimension 72 that admits a large automorphism, where a $d\times d$ matrix is called large, if its minimal polynomial has an irreducible factor of degree $> d/2$.