Throughout this chapter, p is a prime number, and R and F denote the p-adic integer ring ℤpand the p-adic number field ℚp, respectively. We denote by (a) the principal ideal aR for a ∈ F.

An important property of R is that R is a principal ideal domain, that is R is a commutative ring without zero-divisors and every ideal of R is principal. Then it is known that every (finitely generated) module M over R is isomorphic to R/I1 ⊕ … ⊕ R/In where the Ik are ideals of R satisfying I1 ⊂ … ⊂ In which are uniquely determined by M. Therefore if M is torsion-free, i.e. ax = 0 (a ∈ R, x ∈ M) implies a or x = 0, then M has a basis over R.

Let V be a vector space over F. A submodule L over R in V is called a lattice on V if L is finitely generated over R and contains a basis of V over F. Then there exists a basis {vi} of V over F such that L = ⊕iRvi. We can define the discriminant d L of L by det(B(vi, vj)) × (R×)2. This does not conflict with the definition in Chapter 1.

By a regular quadratic lattice over R, we mean a lattice on some regular quadratic space over F. If K is a torsion-free (finitely generated) module over R, then K is a lattice on FK. In this sense, if there is no confusion, we often refer to a submodule M of possibly smaller rank as a regular (sub)lattice when FM is regular.