Universal integral quadratic forms over dyadic local fields

Beli, Constantin N.

doi:10.48550/arxiv.2008.10113

Cited by 2 publications

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“…[2] and [3]). He classified 1-universal lattices over general dyadic fields in [4] by his theory. In a forthcoming work [11], all (classically) k-universal lattices for k ≥ 2 will be determined by using BONGs.…”

Section: On 2-universal Quaternary Lattices Over Dyadic Local Fieldsmentioning

confidence: 99%

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

He¹,

Hu²,

Fei³

2022

Preprint

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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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Section: On 2-universal Quaternary Lattices Over Dyadic Local Fieldsmentioning

confidence: 99%

“…By strong approximation for spin groups, an effective algorithm for determining a universal form over Z is given by the local-global principle. The local universal conditions have been given in [31] and [4]. However, the local-global principle is not always true over general number fields.…”

Section: Introductionmentioning

confidence: 99%

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

He¹,

Hu²,

Fei³

2022

Preprint

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“…The work [HHX22] by Xu and the authors continued the investigation and answered similar questions about k-universality for general k. In the proofs of these results, a key step turns out to be a complete determination of k-universal forms over non-dyadic local fields and some partial results in the dyadic case. For k = 1, Beli's work [Bel20] complements the analysis over dyadic fields in [XZ22, § 2], and gives necessary and sufficient conditions for an integral quadratic form over a general dyadic field to be universal. His method builds upon the general theory of bases of norm generators (BONGs), which he developed in his thesis [Bel01] (see also [Bel06], [Bel10], [Bel19]).…”

Section: Introductionmentioning

confidence: 98%

“…As we have mentioned before, Beli determines all universal lattices over general dyadic fields by using the theory of BONGs in [Bel20]. In the last section of that paper, a translation of his main theorem in terms of Jordan splittings is given without proof.…”

Section: Introductionmentioning

confidence: 99%

“…As far as we can see, to treat that general case, we need a comprehensive knowledge of Beli's theory of BONGs, which in our eyes looks more complicated than the classical theory. Moreover, over ramified dyadic fields we have to state the results in terms of invariants assocaited to BONGs, and unlike the case k = 1 discussed in [Bel20], when k ≥ 2 it seems rather complicated to translate those results in the language of Jordan splittings.…”

Section: Introductionmentioning

confidence: 99%

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On $k$-universal quadratic lattices over unramified dyadic local fields

He,

2022

Preprint

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Let k be a positive integer and let F be a finite unramified extension of Q 2 with ring of integers O F . An integral (resp. classic) quadratic form over O F is called k-universal (resp. classically k-universal) if it represents all integral (resp. classic) quadratic forms of dimension k. In this paper, we provide a complete classification of k-universal and classically k-universal quadratic forms over O F . The results are stated in terms of the fundamental invariants associated to Jordan splittings of quadratic lattices.

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Universal integral quadratic forms over dyadic local fields

Cited by 2 publications

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

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

On $k$-universal quadratic lattices over unramified dyadic local fields

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