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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 3$$\end{document}, we prove that the indefinite k-universal property satisfies the local–global principle over number fields. For k=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document}, 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.