This post started out as an exposition on the monster Lie algebra, but it grew out of control, so I’m hacking off a chunk. Here, I’ll describe the basics of integer lattices.

Lattices show up in many mathematical contexts, some of which may be unexpected to the uninitiated. These contexts include the study of optimal periodic sphere-packings, the topology of 4-manifolds (where lattices give a full classification in the simply connected case), algebraic number theory, finite group theory, and theoretical high-energy physics. I will say almost nothing about these applications, though.

An integer lattice is a free abelian group A of finite rank, equipped with a symmetric integer-valued bilinear form . An isomorphism from one lattice to another is an isomorphism of groups that preserves inner products. There is a rough hierarchy of basic questions:

- How do we classify lattices up to isomorphism?
- How do we tell two non-isomorphic lattices apart?
- What sort of invariants are tractably computable?
- Which lattices are interesting, and what can we say about them?

These questions are not strictly decreasing in generality – we can choose a set of easily computed invariants, and try to classify all lattices for which the invariants are certain quantities. For example, if we wanted to classify lattices of rank 0 or 1, we could determine that the zero lattice is the only lattice of rank zero, and the lattices of rank one are uniquely characterized by a single integer: the inner product of a generator with itself. Here is a lattice where that inner product is a positive integer n, viewed as a subset of the real line:

It is common in treatments of lattices to move from here to a discussion of classifying positive definite rank two lattices, with its connections to orders in imaginary quadratic fields, and the Gauss composition law as a manifestation of the narrow class group. Instead, I will analyze the invariants we’ve already used in a little more depth.

For example, the usual way to consider the rank of a lattice is to forget the inner product and consider the underlying free abelian group, but we can instead forget the integral structure by taking a tensor product with the complex numbers. All inner products on a complex vector space of a fixed dimension and nullspace are equivalent. In other words, this tensor product operation lumps lattices into equivalence classes that are only distinguished by the two invariants that are dimension and nullspace.

This example suggests a general strategy, where we take tensor products of a lattice with various natural choices of rings that are less arithmetically rich than the integers (in particular, the real numbers and p-adic rings). We can then examine the induced bilinear forms, and we get a strong invariant, called the genus. There are many other invariants that I will discuss, but the genus forms a sort of core of the theory.

As another example from the above discussion, the inner product of a generator of a rank one lattice with itself is a measure of how much space lies between lattice vectors. This gives rise to the notion (when the inner product is positive) of covering radius, given by the minimum radius such that setting closed balls of radius centered at each lattice point yields a cover of Euclidean space. An alternative generalization is the determinant (or its square root, depending on your convention) as a measure of the volume of a fundamental parallelotope. Delving further, we can consider the group of reals that have integer inner product with all elements of the lattice, and take its quotient by the lattice. This yields a cyclic group whose order is the determinant, but it is also equipped with a torsion bilinear form. A third generalization comes from creating a generating function that enumerates lattice vectors of a given length. Some Fourier analysis implies this generating function obeys some rather nontrivial functional equations.

The first part of the above paragraph leads to connections with sphere packing and pointwise invariants like kissing number. The alternative generalization suggests we study finite abelian groups equipped with -valued bilinear forms (and these come close to yielding the genus), and the third generalization leads to the theory of theta functions, modular forms, and the Weil representation.

There are informative invariants that are not suggested by the cases of dimension zero and one, such as those arising from symmetries, and I will discuss them at the end of this big list.

**Signature**

Given a lattice , we can consider the vector space with the induced symmetric bilinear form. Sylvester’s law of inertia asserts that his form can be diagonalized, i.e., there exists a basis of such that the inner product of vectors and is given by , where . The multiplicity of -1, 0, and 1 among the is invariant under the choice of diagonalizing basis.

If there are no zeroes, the form is nonsingular, i.e., the maps induced by left and right inputs of the form have trivial kernel. In this case, we can say that , and the lattice has signature , where is the number of positive and is the number of negative $\epsilon_i$. Cases of particular interest are when (resp. ), i.e., when the form is positive (resp. negative) definite. When both p and q are positive (i.e., if the form is indefinite) but at least one of the values is equal to one, then the form is known as a Lorentzian by mathematicians (named after the physicist Lorentz) or Minkowskian by physicists (named after the mathematician Minkowski). Lorentzian lattices and forms appear when studying relativity and hyperbolic geometry, since spacetime has Lorentzian signature, and hyperbolic space is a homogeneous space for the group of Lorentzian isometries.

If the lattice is singular (i.e., if there are vectors in V that are perpendicular to everything including themselves) then the singular subspace of V is spanned by the sublattice , which is a direct summand of L. Singular lattices occasionally show up in mainstream mathematics, e.g., as root lattices of affine Lie algebras.

It is common to refer to signature as the integer r-s rather than the ordered pair (r,s). One can reconstruct the ordered pair from the combination of the integer signature and the rank.

**Even versus odd**

A lattice L is even if all of its vectors have even norm, i.e., if . L is odd if it is not even. While the signature can be viewed as an invariant arising from completing at the infinite place (or the prime -1, according to Conway), the parity is the first step in considering local invariants at the prime 2.

Even lattices have special significance, because they naturally arise from integral quadratic forms. In many cases where 2 is not invertible, quadratic forms appear more naturally than bilinear forms. These include the orders in quadratic fields, MacLane’s calculations of cohomology of , and lattice conformal field theories (where odd lattices yield fermionic fields).

For any signature (r,s), one can construct an odd lattice , with norm given by . These are orthogonal direct sums of one dimensional lattices and . One can also construct the lattice whose coordinates are either all integers or all elements of , with even coordinate sum. These are typically only considered when r is congruent to s mod 8, since those cases yield especially nice properties.

**Determinant**

Given a nondegenerate lattice L, we can construct an invariant by choosing a basis, and taking the determinant of the matrix whose (i,j) entry is the inner product of the i-th basis vector with the j-th basis vector. The matrix is called the Gram matrix of the basis, and the determinant is a rough measure of how loosely packed the lattice vectors are in . Under suitable sign conventions, the determinant is in fact the square of the volume of a fundamental parallelotope of the lattice.

The dual lattice of L is defined by:

Since the bilinear form on L takes values in integers, L is a subgroup of its dual, and its index is finite. The order of the quotient group is the (absolute value of) the determinant. Note that the induced bilinear form on the dual lattice is a priori rational-valued, and it is integer-valued if and only if L is equal to its dual, i.e., if L is unimodular.

The lattices (for all r and s) and (for r and s congruent mod 8) are unimodular, and when both r and s are positive, they form the only odd (respectively, even) unimodular lattices of signature (r,s), up to isomorphism. In the definite case, the situation is quite different, as the number of isomorphism types of definite unimodular lattices has faster than exponential growth in the dimension. Conway and Sloane give a rough heuristic, saying that classification of definite lattices is relatively easy when the sum of the dimension and square root of determinant is at most 24. There has been some progress a little beyond this horizon, but this chart can give you a good idea of how wild the problem becomes.

**Rationalization**

Given a nonsingular lattice L, we can consider , a rational vector space with an inner product. Unlike the real case, we obtain much more information than the signature. In fact, we get a mod 8 invariant called p-signature for each prime p from the tensor product with . For odd primes, this is given by diagonalizing to get a set of p-adic integers , and considering the sum . For 2, one cannot always diagonalize, so the 2-signature is more subtle.

Given a collection of p-signatures (including the real signature) that is trivial for almost all p, there is a mod 8 obstruction to the existence of a rational vector space with a suitable bilinear form that has those signatures. If the obstruction vanishes, then the rational vector space exists and is unique. This gives a classification of lattices up to rational equivalence. When we confine our view to even unimodular lattices, the mod 8 rational obstruction specializes to the requirement that the signature (r,s) satisfy (mod 8).

**Quotient form**

Given a nondegenerate lattice L, the quotient group inherits a -valued bilinear form, and this form yields more information than simply the order of the underlying group, especially if the group has many cyclic factors. If L is even, this bilinear form can be promoted to a -valued quadratic form. These forms are often called discriminant forms, and they encode essentially all of the p-local information of L in a single package (although rank is only determined mod 2 and signature is only determined mod 8).

**Genus**

A genus is an equivalence class of lattices, and the equivalence relation is quite strong. Two lattices are in the same genus if they have the same signature and are isomorphic after taking a tensor product with the p-adic integers for all primes p. We can characterize genera using a notation introduced by Conway. A genus has the form or , where the Roman numeral determines whether the form is even or odd, and (stuff) is a collection of local invariants.

For p an odd prime, one can diagonalize a form to get a direct sum , where is diagonal with entries that are times units. Removing the factors of $p^i$, we can consider whether or not the resulting diagonal matrix of units has square determinant mod p. The data coming from valuations and square class completely classifies forms over p-adic integers for odd p – two p-adic bilinear forms with the same local data are isomorphic by a change of basis. Conway has an abbreviated notation for this invariant: describes a certain 2+5+1+3 = 11 dimensional lattice, where the signs in the exponents are positive for those diagonal factors with square determinant.

For p=2, one cannot diagonalize an arbitrary form, but there is a “2-adic Jordan form”, made of blocks of the form or , where q is a power of 2, a and c are even, and b is odd. The Jordan form is not unique, but gives rise to a set of symbols that distinguish inequivalent forms. One then computes all possible equivalences between symbols.

At the archimedean place, the local invariant is the signature.

**Spinor genus**

There is an extra piece of local data that we can use arising from finer structure in orthogonal groups. Over a field of characteristic not 2, any orthogonal transformation is a composite of reflections in a sequence of hyperplanes perpendicular to vectors of nonzero norm. Even if two p-adic quadratic forms on a vector space are isomorphic, we can consider a decomposition of a given isomorphism into reflections and look at the product of norms of the chosen vectors. Multiplying a vector by a nonzero number preserves the reflection, so we first mod out by the multiplicative group of squares. The resulting square class is called the spinor norm of the transformation. Given a p-adic form f and a square class r, we define to be the set of all p-adic forms g such that there exists a transformation from f to g of determinant 1 and spinor norm r. is the p-adic spinor genus of f. The set of r such that is a subgroup of square classes called the spinor kernel of f. For p relatively prime to the determinant of f, the spinor kernel is the full group of square classes, so each genus gets split into finitely many spinor genera.

Eichler proved that for indefinite forms of rank at least 3, a spinor genus contains exactly one integral equivalence class. In other words, the spinor genus of an indefinite lattice of rank at least 3 uniquely determines its isomorphism type.

**Root system**

A root of a lattice L is an element such that reflection by its orthogonal hyperplane in yields an automorphism of the lattice. Any vector of norm is automatically a root. The positive definite lattices generated by roots have been classified, and they form an essential part of the classification of semisimple complex Lie algebras. The reflections in roots generate an important subgroup of the automorphism group of a lattice, and when the lattice is indefinite, one often obtains an arithmetically interesting infinite group.

**Automorphism group**

The group of automorphisms of a lattice is useful not only as a way to distinguish different lattices, but as a way to enumerate positive definite lattices of a given genus. This is because the Smith-Minkowski-Siegel mass formula gives a numerical value for the total number of isomorphism classes, weighted by automorphism group. If you make a list of mutually nonisomorphic lattices, and find that the sum of the reciprocals of the orders of their automorphism groups is equal to the value you expect, then you know that you’ve given a complete enumeration. This was used by Niemeier (and possibly Witt) when classifying 24 dimensional even unimodular lattices. More recently, my academic brother Oliver King used it to investigate the number of isomorphism types of even unimodular lattices of rank 32 with no roots. Some number theorists had speculated that this class of lattices might be small enough that one could reasonably study them as individuals, but King gave a lower bound of 10 million isomorphism types.

Many positive definite lattices (in small dimension) yield interesting automorphism groups, and we can view them as groups of integral points in unusual forms of orthogonal groups. The root lattices for classical groups are symmetric groups or extensions of symmetric groups by elementary abelian 2-groups, and the root lattices for the exceptional groups , , and have orientation-preserving automorphisms given by simple groups of order 25920, 1451520, and 348364800, respectively. The Leech lattice (which is the unique even unimodular lattice in dimension 24 with no roots) has automorphism group Co0, which has order 8315553613086720000 and is a degree 2 central extension of Conway’s sporadic simple group Co1. The absence of roots means there are no obvious reflection symmetries in the lattice, and in fact Co0 has no nontrivial maps to . There are several other sporadic simple groups that appear here: Co2 is the stabilizer of a norm 4 vector, Co3 is the stabilizer of a norm 6 vector, McL is the stabilizer of a 4-4-6 triangle or a certain norm 16 vector, HS is the stabilizer of a 4-6-6 triangle or a norm 14 vector, and the Mathieu groups also arise as stabilizers.

**Theta function**

Given a positive definite integer lattice L of rank r, you can form a generating function , which is a priori a formal power series in , but the coefficients grow slowly enough that it forms a holomorphic function on the complex upper half plane. If L is even, the theta function is periodic of period 1, while if L is odd, it has period 2.

exhibits additional symmetry, arising from Poisson summation, and is a modular form of weight r/2 for some finite level congruence subgroup of . In other words, we can view it as a section of some line bundle on a parameter space of structured elliptic curves, at least when r is a multiple of 4. Since these modular forms often form vector spaces of small dimension, we can constrain properties of lattices by doing computations with spaces of modular forms. For example, the space of level one modular forms is zero for weights between zero and four. If L is even and unimodular, then has level one. This implies there are no positive definite even unimodular lattices of dimension less than eight. We can also eliminate dimension 12 (resp. 20), since the space of weight 6 (resp. 10) forms is spanned by a form with both positive and negative coefficients. In dimension 8, the lattice has theta function equal to the Eisenstein series , because the space of forms of weight 4 is one dimensional. Similarly, the even unimodular lattices and have the same theta functions, since the space of level 1 forms of weight 8 is one dimensional. This fact played an integral role in Milnor’s solution to Mark Kac’s question about the spectra of Laplacians on manifolds (cf. “can one hear the shape of a drum?”).

One can combine theta functions corresponding to cosets of a lattice in its dual to make a vector-valued theta function. This yields Weil’s representation of on for even rank, and a representation of the double cover for odd rank.

For the case of indefinite lattices, there is a complication, since the naive generating function will yield a divergent sum for many powers of q. One can still construct theta functions, but they are substantially more complicated, as they involve nonholomorphic contributions and depend on an auxiliary variable that lives on the noncompact Grassmannian . These functions may reappear later in this series when I actually talk about Lie algebras.

Standard references: Conway, Sloane *Sphere packings, lattices and groups* and Conway, Fung *The sensual (quadratic) form.*

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Very nice post. Thanks!

One comment: discriminant forms encode all the “integral” p-adic information as well; If two lattices have isometric discriminant forms and same signature, then they are in the same genus.

Maybe that’s what you meant by “p-local information”, but the ordering of your presentation might suggest that only the “rational” p-adic invariants can be recovered from the discriminant form…