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

integralityof the metric on a lattice andwhy? In the case of complex elliptic curves, does it have to do with the resulting curves being defined over $\mathbb{Q}$?