In this chapter, we investigate functorial properties of positive definite quadratic forms with respect to the tensor product and scalar extension.

Let L, M and N be quadratic modules over ℤ and RK the maximal order of an algebraic number field K. Some of the fundamental problems are:

When we consider these in the category containing indefinite quadratic forms, it is not interesting. For example, let M, N be unimodular positive definite quadratic modules over ℤ with the same rank and n(M) = n(N) = 2ℤ, then L ⊗ M ≅ L ⊗ N holds for L = 〈1〉 ⊥ 〈-1〉. So the condition L ⊗ M ≅ L ⊗ N says nothing. If the field K is not totally real, RKM is isotropic at an infinite imaginary place of K and hence the isometry class of RKM is nothing but its spinor genus (by the generalization of Theorem 6.3.2). Therefore RKM ≅ RKN holds. So the above problems are meaningless.

However if we confine ourselves to the category of positive definite quadratic forms and totally real algebraic number fields, the answer seems affirmative. At least there is no counter-example so far. Let us give some results in this chapter.

In this chapter, by a positive lattice we mean a lattice on a positive definite quadratic space over. Hence, if L is a positive lattice, then for a basis {ei} of L over ℤ, B(ei, ej) ∈ and the matrix (B(ei, ej)) is positive definite.