On Automorphisms of Extremal Even Unimodular Lattices

Abstract

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with ϕ(m) > 24. In particular the lattice P 48n is the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

A fourth extremal even unimodular lattice of dimension 48

“…is not too small. In [12] I classified all 48-dimensional extremal lattices that have an automorphism of order a whose Euler phi value is ϕ(a) > 24. All these lattices are isometric to one of the lattices P 48p , P 48q , or P 48n , which were known before.…”

“…It turns out that there is a unique such lattice, P 48m , and this lattice is not isometric to one of the lattices above. (SL 2 (25) × PSL 2 (7)) : 2 5241600 = 2 8 3 2 5 2 7 13 [11], [12] 2 The type of an automorphism…”

Automorphisms of extremal unimodular lattices in dimension 72

“…So If p divides ℓ we also put d 1 (p) := dim Fp ((L #,p ) 1 /L 1 ) and d ζ (p) := dim Fp ((L #,p ) ζ /L ζ ). Then we have for any prime divisor q of ℓ that q d 1 (q)+d ζ (q) = |L #,q /L| is the q-part of the determinant of L (for p = q this is investigated in more detail in [18]).…”

“…The main notion to deal with automorphisms is the one of the type of an automorphism σ of prime order p of the lattice L introduced in [16] (see Section 5.1). It is independent of the quadratic form on L and determines the Z p [σ]-module structure of Z p ⊗ Z L.…”

Some congruences for Siegel theta series

“…In this section we recall and modify some results from [6,11] (see also [9,Section 4]) and fix some notations for the rest of the article.…”