Representations of positive definite Hermitian forms with approximation and primitive properties (Q1323860)

Let \(M\) be a positive definite Hermitian lattice of rank \(m \geq 2n+1\) and \(T\) be a finite set of primes containing the prime divisors of 2 vol \((M)\). If \(N\) is a Hermitian lattice of rank \(n\) which is locally representable by \(M\) and \(\min (N)\) is sufficiently large, then there exists a global representation \(f:N \to M\) such that \(f\) approximates the given local representation at \(T\) and \(f(N_ p)\) is primitive in \(M_ p\) for all primes \(p \not \in T \cup \{q\}\) where \(q\) is any fixed prime not in \(T\). The proof parallels the arithmetic arguments for quadratic forms in \textit{J. S. Hsia}, \textit{Y. Kitaoha} and \textit{M. Kneser} [J. Reine Angew. Math. 301, 132-141 (1978; Zbl 0374.10013)], with some refinements from a recent paper of \textit{M. Jöchner} and \textit{Y. Kitaoka} [``Representations of positive definite quadratic forms with congruence and primitive conditions'', J. Number Theory (1994)].