History of Lattices (reviewed)

Integral lattices have a deep connection to the objects in the LMFDB in that every lattice, $L$, has an associated theta series, $\Theta_L$, which is a holomorphic modular form of half-integral weight. Specifically, the theta series are a type of Siegel modular form, first introduced by Siegel in the 1930's [MR:0332663]. Later, in 1973, Shimura developed a partial correspondence lifting half-integral weight theta series to full integral weight series [MR:0332663], allowing lattices to be studied using the full analytic machinery of modular forms.

The Representation Problem

The lattice $L$, with associated quadratic map $Q$, is said to represent an integer $n$ if the equation $Q(v)=n$ is solvable for $v$ in $L$. The representation problem, or the determination of the number of ways a certain integer can be represented by a given lattice, is one of the classical problems in integral lattices. To this end, the theta series is a valuable tool, in that the $n^{th}$ Fourier coefficients of $\Theta_L$ counts the number of representations of $n$ by $L$.

Classically, sums of squares are one interesting class of lattices; the sums of two squares (2.1.2.1.1), three squares (3.1.2.1.1), and four squares (4.1.2.1.1) all appear in the database. These lattices can all be considered as discrete subsets of Euclidean space, and their associated bilinear map is just the standard inner product. Originally studied by Euler, Legendre, Gauss, Lagrange, and Jacobi, the number of ways an integer can be represented by sums of two, three, or four squares is well known.

In the 1920's, Hasse proved that an integer is represented by a quadratic form provided that it is represented locally by the form at every prime. In the case of indefinite lattices there is more or less an integral analogue of Hasse's result described by Hsai, Shao, and Xu in [MR:1604472]. In the definite case a direct integral analogue of Hasse's local-global result does not exists for lattices, although in several cases there is an asymptotic version; a summary of these results is given by Schulze-Pillot in [MR:2060206].

Given the Siegel correspondence, many of the results in the preceding paragraph rely on a careful melding of the analytic machinery of theta series, and the algebraic techniques of lattices.

Lattices and Codes

There is also an important connection between lattices and codes described by Ebeling in [MR:1938666]. Given a finite commutative ring, $R$, a code, $C$, is an $R$-submodule of $R^n$. A code $C$ is related uniquely to a lattice $L_C$ via Construction A, described by Nebe, Rains, and Sloane in [MR:2209183]. Many of the concepts of lattices theory have analogues in coding theory, for example the theta series of a lattice is a modular form invariant under the action of a certain Siegel modular group, this corresponds directly to the weight enumerator of a code which is a polynomial invariant under the action of the so-called Clifford-Weil group.

As an example, the 8-dimensional root lattice $E_8$ (8.1.1.1.1) corresponds to the well-known Hamming code, an error-correcting code over the finite field $\mathbb{F}_2$. In addition, the 24-dimensional Leech lattice (24.1.1.24.1) corresponds to the Golay code.