The number of embeddings of quadratic \(\mathbb{Z}\)-lattices (Q1914020)

This paper is a continuation of previous work of the author [Proc. Symp. Pure Math. 58, Part 2, 265-274 (1995; Zbl 0827.11018)] on the number of inequivalent primitive embeddings of one quadratic lattice into another, modulo the action of the orthogonal group (or various special subgroups). Specifically, let \(L\) be a unimodular lattice on an \(S\)-indefinite quadratic space \(V\) of dimension \(n\geq 3\) over an algebraic number field \(F\), where \(S\) is a Dedekind set of spots on \(F\). For a second \(S\)-lattice on a nondegenerate quadratic space over \(F\) with dimension not exceeding \(n\), let \(N(L, M)\) and \(N'(L,M)\) denote the numbers of inequivalent primitive representations of \(M\) by \(L\), modulo the action of the orthogonal group \(O(L)\) and the spinorial kernel \(O' (L)\), respectively. In a previous paper [ibid.] the author proved that under certain conditions on local Witt indices at the real spots on \(F\), the computation of \(N' (L, M)\) is reduced via a product formula to the determination of corresponding local indices \(e_p= N' (L_p, M_p)\) for various completions with respect to finite prime spots \(p\) on \(F\). Further global results were then obtained by considering the action of the quotient group \(O(L)/ O' (L)\). The present paper is primarily devoted to completing the computation of the local index \(N' (L_2, M_2)\) for the case when 2 is a prime element in \(F\) (i.e., the dyadic unramified case) and \(L\) is an even lattice. As a consequence, all situations where there is a unique global embedding modulo the action of \(O(L)\) are determined for even \(\mathbb{Z}\)-lattices \(L\). The results obtained are compared with those announced by \textit{R. Miranda} and \textit{D. R. Morrison} [Proc. Japan Acad., Ser. A 61, 317-320 (1985; Zbl 0589.10020); ibid. 62, 29-32 (1986; Zbl 0594.10012)].