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- Title
On indefinite k-universal integral quadratic forms over number fields.
- Authors
He, Zilong; Hu, Yong; Xu, Fei
- 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 ≥ 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.
- Subjects
QUADRATIC forms; INTEGRALS; QUADRATIC fields; QUADRATIC equations; INTEGRAL representations
- Publication
Mathematische Zeitschrift, 2023, Vol 304, Issue 1, p1
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-023-03280-z