The Distribution of Lattices Arising From Orders In Low Degree Number Fields

Orders in number fields provide interesting examples of lattices. We ask: how are lattices arising from orders in number fields distributed? An order O of absolute discriminant Δ in a degree n number field has n successive minima 1 = λ₀ ≤ λ₁ ≤ ≤ λₙ₋₁. For 3 ≤ n ≤ 5 and many G ⊆ Sₙ, we compute the distribution of the points (logΔλ₁,,logΔλₙ₋₁) ∈ Rⁿ⁻¹ as O ranges across orders in degree n fields with Galois group G as Δ → ∞. In many cases, we find that the distribution is given by a piecewise linear expression and is supported on a finite union of polytopes.