Remark

Let $L$ be a free lattice of $V$ with respect to $A$. Since $A_p$ is a principal ideal domain for $p \in P$, $N(p)$ is a free $A$-module of the same rank as $L$ and there exists $u(p) \in \mathbf{GL}(V)$ such that $u(p)(L_p) = N(p)$; this condition moreover determines $u(p)$ to within right multiplication by an element of $\mathbf{GL}(L_p)$. The condition $N(p) = L_p$ for almost all $p \in P$ means that necessarily $u(p) \in \mathbf{GL}(L_p)$ for almost all $p \in P$. The families $(u(p))_{p \in P}$ satisfying the latter property form a multiplicative group $\mathbf{GL}_a(V)$ containing as subgroup the product $\prod_{p \in P} \mathbf{GL}(L_p)$. Theorem 3 then shows that the set of reflexive lattices of V is in canonical one-to-one correspondence with the homogeneous space $\mathbf{GL}_a(V)/\prod_{p \in P} \mathbf{GL}(L_p)$. If a basis $(e_i)_{1 \leq i \leq n}$ of L over A is chosen, $\mathbf{GL}(V)$ (resp. $\mathbf{GL}(L_p)$) is identified with the group of invertible matrices $\mathbf{GL}(n, K)$ (resp. $\mathbf{GL}(n, A, )$) and the group $\mathbf{GL}_a(V)$ with the group of systems of matrices of order $n$, $(U(p))_{p \in P}$, such that $U(p) \in \mathbf{GL}(n, K)$ for all $p \in P$ and $U(p) \in \mathbf{GL}(n, A_p)$ for almost all $p \in P$. If A is a Dedekind domain, the group $\mathbf{GL}_a(V)$ is also identified with the group $\mathbf{GL}(n, A)$, where A is the ring of restricted adèles ($§ 2$, no. 4).