Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$
For $n\ge 3$, confirming a weak version of a conjecture of Hoffmann, we show that every anisotropic quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite extension of the base field of degree not divisible by $4$. The first new case is $n=4$, where we obtain a classification of the...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.668/ |
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| Summary: | For $n\ge 3$, confirming a weak version of a conjecture of Hoffmann, we show that every anisotropic quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite extension of the base field of degree not divisible by $4$. The first new case is $n=4$, where we obtain a classification of the corresponding quadratic forms up to odd degree base field extensions and get this way a strong upper bound on their essential $2$-dimension. As well, we compute the reduced Chow group of the maximal orthogonal grassmannian of the quadratic form and conclude that its canonical $2$-dimension is $2^n+2^{n-2}-2$. |
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| ISSN: | 1778-3569 |