ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11, 94 Roegner, Katherine, Technische Universität Berlin, Germany Computing even, unimodular lattices with complete root systems Let Λ ⊂ ℝ^{n} be an integral lattice (i. e., a ℤ-module of rank n such that ⟨x, y⟩ ∈ ℤ for all x, y ∈ Λ, where ⟨ , ⟩ denotes the usual Euclidean product) with basis λ_{1}, …, λ_{n}. Let S be the n × n matrix consisting of entries s_{ij} = ⟨λ_{i}, λ_{j}⟩. Λ is called unimodular if |det S| = 1. It is said to be even if ⟨x, x⟩ ∈ 2ℤ for all x ∈ Λ. The root system R_{Λ} of Λ, given by

R_{Λ} = {x ∈ Λ | ⟨x, x⟩ = 2},

is complete if [Λ : R_{Λ}] < ∞. Techniques for computing a particular class of even, unimodular lattices with complete root systems are discussed.