On indefinite $k$-universal integral quadratic forms over number fields

Let k ≥ 1 be an integer. An integral quadratic form is called k-universal if it represents all integral quadratic forms of dimension k. We prove that the k-universal property satisfies the local-global principle over number fields for k ≥ 3. We also show that a number field F admits an integral quadratic form which is locally 2-universal but not globally if and only if the class number of F is even. When it is the case, there are only finitely many classes of such forms over F . A variant for classic ternary forms is also discussed. In particular, we classify all the quadratic fields over which there exist classic ternary quadratic forms that are locally universal but not globally.

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