Extensions of representations of integral quadratic forms

Let N and M be quadratic Z-lattices, and K be a sublattice of N. A representation sigma : K -> M is said to be extensible to N if there exists a representation rho : N -> M such that rho vertical bar (K) =sigma. We prove in this paper a local- global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite Z-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132-141, 1978) and Jochner and Kitaoka (J. Number Theory 48:88-101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed.