The number of embeddings of hermitian lattices
Let \(F\) be an algebraic number field, \(K\) a quadratic extension of \(F\), \(V\) a nondegenerate indefinite hermitian space over \(K\) of dimension \(n\) at least 3, and \(L\) a unimodular lattice on \(V\). In this paper, the author determines the number of primitive embeddings of a hermitian lattice \(M\) into \(L\) which are inequivalent modulo the action of the special unitary group \(\text{SU} (L)\). This primitive embedding problem is first solved locally. Global results then follow by applying the strong approximation theorem of \textit{G. Shimura} [Ann. Math., II. Ser. 79, 369-409 (1964; Zbl 0144.295)]. A product formula for the number of global primitive embeddings is obtained when the space \(V\perp (-W)\) has local Witt index at least \(m+1\) when localized at some infinite prime spot on \(F\), where \(W\) is the ambient space for \(M\) and \(m\) is its dimension. The criteria for local primitive representability are easily described for the localization with respect to a prime \(p\) of \(F\) which is not a dyadic prime ramified in \(K\). For such a localization, write \(M_p= M_p (0) \perp M_p (1)\), where \(M_p (0)\) is unimodular and the scale of \(M_p (1)\) contains no units, and let \(m_p (1)\) be the rank of \(M_p (1)\). Then \(L_p\) primitively represents \(M_p\) if and only if either \(m+ m_p (1) <n\), or \(m+ m_p (1) =n\) and \(dM_p (0)= (-1)^{n-m} dL_p\), where \(dM_p (0)\) and \(dL_p\) denote the respective discriminants. The criteria are predictably more complicated in the case of ramified dyadic primes. The analogous primitive embedding problem for unimodular quadratic lattices has been treated in previous papers by the author [J. Number Theory 44, 356-366 (1993; Zbl 0781.11017); Math. Z. 215, 465-475 (1994; Zbl 0790.11028); and Proc. Symp. Pure Math. 58, 265-274 (1995; Zbl 0827.11018)].
scientific article; zbMATH DE number 815912