On indefinite k-universal integral quadratic forms over number fields

An integral quadratic lattice is called indefinite k-universal if it represents all integral quadratic lattices of rank k for a given positive integer k. For k = 3, we prove that the indefinite k universal property satisfies the local-global principle over number fields. For k = 2, we show that a number field F admits an integral quadratic lattice which is locally 2-universal but not indefinite 2-universal if and only if the class number of F is even. Moreover, there are only finitely many classes of such lattices over F. For k = 1, we prove that F admits a classic integral lattice which is locally classic 1-universal but not classic indefinite 1-universal if and only if F has a quadratic unramified extension where all dyadic primes of F split completely. In this case, there are infinitely many classes of such lattices over F. All quadratic fields with this property are determined.