CHAPTER 1 Preliminaries Let n 1 be a natural number and let A be a free Z-module of rank n. We call a submodule of A a lattice if it is a free Z-module of rank n. If L is a lattice of A, then we have (A : L) oo. We denote by £ the set of all lattices of A. For any prime number p, put Ap = A ®z Zp. Then ^4P is a free Zp-module of rank n. We call a Zp-submodule of Ap a Zp-lattice if it is a free Zp-module of rank n. If Lp is a Zp-lattice of Ap, then (Ap : Lp) = p171 for some integer m 0. We denote by £ p the set of all Zp-lattices of Ap. Further we denote by £/ the subset of the direct product Ylp &p consisting of elements all but a finite number of whose components coincide with Ap. Here the product extends over all prime numbers. The following lemma is well known. LEMMA 1.1. The mapping defined by £ 3 L i— {Lp)p G £/, Lp = L 0 Z Zp is a bijection satisfying (A : L) = Ylp(Ap : Lp). Now we assume that A is also a commutative associative ring with 1. We call a lattice of A an order if it is a subring of A containing 1. Similarly we call a Zp-lattice of Ap a Zp- order if it is a subring of Ap containing 1. We denote by 31 and CRP the set of all orders of A and the set of all Zp-orders of Ap, respectively. Then we have X c i L and $ p C £ p . We put K^-C'nJJOip. p By restricting the bijection in Lemma 1.1, we have LEMMA 1.2. The mapping defined by S ^ O ^ (Op)p G # , 0 P = 0 g z Zp 25 a bijection satisfying (A : O) = Ylp(AP : Op). We introduce the following Dirichlet series: oeR opeJip By Lemma 1.2, we have PROPOSITION 1.1. VA{S) = Y[VAAS)- P 5

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