Pfister forms and a conjecture due to Colliot–Thelène in the mixed characteristic case
- Authors: Panin I.A.1, Tyurin D.N.1,2
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Affiliations:
- St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
- Leonard Euler International Mathematical Institute at Saint Petersburg (SPB LEIMI)
- Issue: Vol 88, No 5 (2024)
- Pages: 174-186
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/265540
- DOI: https://doi.org/10.4213/im9566
- ID: 265540
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Abstract
Let $R$ be a regular local ring of mixed characteristic $(0,p)$, where $p\neq 2$ is a prime number.Suppose that the quotient ring $R/pR$ is also regular. We fix a non-degenerate Pfister form $Q(T_{1},\ldots,T_{2^{m}})$ over $R$and an invertible element $c$ in $R$. Then the equation $Q(T_{1},\ldots,T_{2^{m}})=c$ has a solution over $R$if and only if it has a solution over the fraction field $K$.
About the authors
Ivan Alexandrovich Panin
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Email: paniniv@gmail.com
Doctor of physico-mathematical sciences
Dimitrii Nikolaevich Tyurin
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences; Leonard Euler International Mathematical Institute at Saint Petersburg (SPB LEIMI)
Scopus Author ID: 57196744354
without scientific degree
References
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- J.-L. Colliot-Thelène, “Formes quadratiques sur les anneaux semi-locaux reguliers”, Colloque sur les formes quadratiques, 2 (Montpellier, 1977), Bull. Soc. Math. France Mem., 59, 1979, 13–31
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- S. Scully, “The Artin–Springer theorem for quadratic forms over semi-local rings with finite residue fields”, Proc. Amer. Math. Soc., 146:1 (2018), 1–13
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