Famous ideal lattices

Question

I am wondering if there exist some special rings $R$ that gives us, under the canonical embedding, some special lattices, like the root lattices, Barnes-Wall lattices, Leech lattices, ...

In more details, is there some rings $R$ such that $\sigma(R)=D_n, \, A_n, \, BW^n ...$?

Answer 1

Yes, there are. The following table is taken from this paper of Ducas and van Woerden, although the results are not derived there (in the below, $p$ is an odd prime, and $n, m$ are coprime). \begin{align*} \mathbb{Z}[\zeta_{2^k}]&\cong \mathbb{Z}^{2^{k-1}}\\ \mathbb{Z}[\zeta_p] &\cong A_{p-1}^*\\ \mathbb{Z}[\zeta_{p^k}]&\cong \bigoplus_{i = 1}^{p^k-1}\mathbb{Z}[\zeta_p]\cong I_{p^k\times p^k}\otimes \mathbb{Z}[\zeta_p]\\ \mathbb{Z}[\zeta_{nm}]&\cong \mathbb{Z}[\zeta_n]\otimes \mathbb{Z}[\zeta_n] \end{align*} Conway and Sloane's book Sphere Packings, Lattices, and Groups is the canonical reference on the topic of lattices. Chapter 8 discusses "algebraic" constructions of lattices. This means two things:

1. Constructions of lattices as $R$-modules of a certain rank for $R\neq\mathbb{Z}$
2. Constructions of lattices as ideals within (the ring of integers of) an algebraic number field

Note that these things are slightly different. For example, an ideal lattice has rank at most the degree of the underlying number field (as it is a sublattice of $\mathcal{O}_K$), but there is no such restriction on the first case.

Anyway, chapter 8, section 7.3 gives a general equation for the lattice $\mathbb{Z}[\zeta_m]$, in particular its Gram matrix $A$ has $(j, k)$ entry: $$A_{j, k} = \frac{\mu(d)\varphi(m)}{\varphi(d)},\quad d = \frac{m}{(m, k-j)}$$

They then summarize constructions of $E_6$ as an ideal of $\mathbb{Z}[\zeta_9]$, the Leech lattice as an ideal of $\mathbb{Z}[\zeta_{39}]$ (note that $\varphi(39) = 24$).

There are a few other constructions given as well, but those are the "biggest names".

Edit:

One can further realize Craig's Lattices as ideals (again within the ring of integers of a cyclotomic number field). Craig's Lattices are a family of lattices $A_n^{(m)}$ obtained by starting with $A_n^{(0)} = \mathbb{Z}^{n+1}$, and performing a "repeated differencing" operation. I will not define the construction here, but notably:

1. $A_n^{(1)}= A_n$ is the (primal) $A_n$ root lattice.
2. According to Conway and Sloane, Craig's lattices are the densest packings known for dimension $148 \leq n \leq 3000$. I do not know how/if this knowledge has changed in the last 20 years.

One can then obtain $A_{p-1}^{(m)}$ as the ideal $(1-\zeta_p)^m$ within the ring of integers of $\mathbb{Z}[\zeta_p]$. The minimal norm (which I believe Conway and Sloane uses to denote the squared norm) is not explicitly known, but can be lower-bounded (in the case that $n = p-1$, and $m < n/2$) by $2m$. See chapter 8 section 6 of Conway and Sloane, or Proposition 5.4.7 of Martinet's Perfect Lattices in Euclidean Space.