Automorphisms of extremal unimodular lattices in dimension 72

Abstract

The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in Magma[7] 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 Γ72 from [17] is the unique extremal even unimodular lattice of dimension 72 that admits a large automorphism, where a d×d matrix is called large, if its minimal polynomial has an irreducible factor of degree >d/2.