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Representations of positive definite Hermitian forms with approximation and primitive properties. (English) Zbl 0797.11040
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 J. S. Hsia, Y. Kitaoha and M. Kneser [J. Reine Angew. Math. 301, 132-141 (1978; Zbl 0374.10013)], with some refinements from a recent paper of M. Jöchner and Y. Kitaoka [“Representations of positive definite quadratic forms with congruence and primitive conditions”, J. Number Theory (1994)].

MSC:
 1.1e+40 Bilinear and Hermitian forms
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