Lattices: why require bilinear form to be integral?

Question

This is a quite localized question, but I hope it won't be closed as unfit to MO. Well, a lattice $\Lambda$ in $\mathbb{R}^n$ is a discrete subgroup generated by a basis. Such a lattice gets a positive definite symmetric bilinear (p.d.s.b.) form $\Lambda\times\Lambda\rightarrow\mathbb{R}$ by restriction from the standard Euclidean form, and we can speak of the isometry type of $\Lambda$. One can of course do this abstractly, defining a lattice to be a finite rank abelian group $\Lambda$ together with a p.d.s.b. form $q$ on $\Lambda\otimes _{\mathbb{Z}}\mathbb{R}$ (or maybe a real-valued p.d.s.b.f. $q: \Lambda\times\Lambda\rightarrow\mathbb{R}$).

I came across the wikipedia page on unimodular lattices and the page on isometries of Euclidean space.

Well, it is my impression (I'm no expert in lattices) that in theories such as the one about "space groups" there is no need of requiring that the metric $q: \Lambda\times\Lambda\rightarrow\mathbb{R}$ be integer valued, because any geometric situation can arise as a relization of a $\Lambda$ inside $\mathbb{R}^n$. Also, in the "elementary" theory of complex elliptic curves, any lattice can give rise to one, even if the restriction of the canonical metric on $\mathbb{C}$ to the lattice points is not integer-valued. On the other hand, in the wikipedia page on unimodular groups it is required in the definition that the form be integer-valued. Also, in the theory of complex toruses $\mathbb{C}^n/\Lambda$, the presence of an integral Riemann bilinear form on $\Lambda\otimes _{\mathbb{Z}}\mathbb{R}$ implies projectivity. So, my question simply is:

In which contexts is it natural to impose integrality of the metric on a lattice and why? In the case of complex elliptic curves, does it have to do with the resulting curves being defined over $\mathbb{Q}$?

Answer 1

One problem that seems to be implicit in your question is that the term "lattice" is used in many contexts, and has multiple definitions. Among people who work with integral bilinear forms or quadratic forms, the norm on a lattice is defined to take values in integers, but it is definitely not assumed to be positive definite (contrary to the definition you offered at the top).

The most classical arithmetic origins come from Brahmagupta's 7th century work on binary quadratic forms, leading to Gauss's composition law and questions related to Waring's problem about which natural numbers are represented by quadratic forms of a certain type. As others have mentioned, there are applications that arise quite naturally in studying the cohomology (in particular, intersection theory) of manifolds, and in number theory proper, where they arise in the study of modular forms via theta constants and trace forms. If you want to study orthogonal groups or Clifford algebras in a setting that includes both real coefficients and finite fields, it is necessary to consider quadratic forms defined over number rings, and in particular, lattices that take values in the integers. A more recent application is in lattice conformal field theory, where you need the bilinear form on a lattice to be integer-valued to yield a super vector space of states (and it must be even to yield an honest vector space).

In the case of complex elliptic curves, the integrality of the lattice (after suitable rescaling) is equivalent to the elliptic curve having complex multiplication. There are plenty of interesting things to say about how these curves relate to class field theory, but it doesn't have much to do with the curves being defined over $\mathbb{Q}$. Most curves over $\mathbb{Q}$ do not have CM, and most CM curves are not defined over $\mathbb{Q}$. See chapter 2 in Silverman's Advanced Topics.

Answer 2

One place integral bilinear forms are important in topology, because for an even-dimensional manifold, the cohomology in half the dimension has a unimodular integral bilinear form: cup product maps it to the n-dimensional cohomology, which (for a closed oriented manifold) is $\mathbb Z$. For dimensions of the form $4n+2$, the forms are antisymmetric and all such forms are isomorophic, but for dimensions of the form $4n$, they give important information.

As Mariano Suárez-Alvarez says in comments, integer-valued forms often arise in the course of thinking about many kinds of mathematics; this is often because algebraic numbers are explicitly or implicitly involved, as in Qiaochu Yuan's answer. One very common way this happens is that if something is described by an integer matrix $M$, then any minimal invariant rational subspace for $M$ has coordinates identifying it with the field $\mathbb Q(\lambda)$, where $\lambda$ is any one of the real or complex eigenvalues of $M$, and where all integer lattice points map to algebraic integers. So there's the same integer-valued bilinear form in the background as in Qiaochu Yuan's answer.

Answer 3

In algebraic number theory, the bilinear forms we care about are trace forms on rings of integers $\mathcal{O}_K$ of number fields $K$, and these are automatically integral. This is important because their reduction modulo $p$ contains e.g. information about ramification.