In this chapter and the next, we study quadratic forms over ℤ, which is the main goal of this book. All previous chapters converge here. There are still many facts which we do not cover in this chapter. The next has another character.

Fundamentals

Here we give some basic notations and facts. First of all we give a correspondence between local and global lattices.

Let a field F and a ring R be either ℚ and ℤ or ℚp and ℤp. A module L over R in a finite-dimensional vector space V over F is called a lattice on V if L is finitely generated and contains a basis of V. Then L has a basis over R and rank L = dim V.

Let V be a regular quadratic space over. A lattice on it is called a regular (quadratic) lattice.

If V∞ is positive definite, i.e. Q(x) > 0 for every non-zero x ∈ V∞, then V is called positive definite and a lattice on it is called a positive (definite quadratic) lattice. If V∞ is negative definite, i.e. Q(x) < 0 for every nonzero x ∈ V∞, then V is called negative definite and a lattice on it is called a negative (definite quadratic) lattice.

If V∞ is isotropic, then V is called indefinite and a lattice on it is called an indefinite (quadratic) lattice. We note that if V is indefinite and dim V ≥ 5, then V is isotropic by Theorem 4.4.1 and 3.5.1.