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arXiv · 2201.10730

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

Zilong He·Yong Hu·Fei Xu
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Abstract

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 $k\geq 3$, 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.

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Zilong He, Yong Hu, Fei Xu. 2022-01-26. On indefinite $k$-universal integral quadratic forms over number fields. https://doi.org/10.1007/s00209-023-03280-z

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