The number of embeddings of hermitian lattices
type: Item
instance of: scholarly article
title: The number of embeddings of hermitian lattices
zbMATH Open document ID: 0834.11017
author: Donald G. James
published in: Journal of Algebra
publication date: 1996-03-31T00:00:00Z
review text: 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)].
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11E08
zbMATH DE Number: 815912
zbMATH Keywords: hermitian form
zbMATH Keywords: unimodular lattice
zbMATH Keywords: primitive embeddings
zbMATH Keywords: hermitian lattice
zbMATH Keywords: special unitary group
MaRDI profile type: Publication
full work available at URL: jabr.1995.1246
OpenAlex ID: W2077126000
DOI: 10.1006/JABR.1995.1246
description: scientific article; zbMATH DE number 815912