On the reduced unramified Witt group of the product of two conics

IF 0.5 4区 数学 Q3 MATHEMATICS
Alexander S. Sivatski
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引用次数: 0

Abstract

We investigate the reduced unramified Witt group of the product of two smooth projective conics \(X_1\), \(X_2\) over a field. In some particular cases, this group denoted as \(W(X_1,X_2)\) turns out to be very small (zero or \({\mathbb {Z}}/2{\mathbb {Z}}\)). On the other hand, certain examples when it is infinite are constructed. We give sufficient conditions providing nontriviality of \(W(X_1,X_2)\) in terms of 2-fold Pfister forms \(\pi _1\), \(\pi _2\) associated with the conics. These conditions and constructions of the corresponding nonzero elements in \(W(X_1,X_2)\) depend on \({\text {ind}}(\pi _1+\pi _2)\). Also we study the question of triviality (nontriviality) of this group with respect to extensions of the ground field.

Abstract Image

论两个圆锥的乘积的还原无ramified 维特群
我们研究了域上两个光滑投影圆锥 \(X_1\),\(X_2\)的乘积的还原无ramified Witt 群。在某些特殊情况下,这个表示为 \(W(X_1,X_2)\) 的群很小(零或 \({\mathbb{Z}}/2{mathbb{Z}}\))。另一方面,我们也构造了一些当它无限大时的例子。我们给出了提供与圆锥相关的 2 折普菲斯特形式 \(\pi _1\), \(\pi _2\) 的 \(W(X_1,X_2)\)的非难性的充分条件。这些条件和在\(W(X_1,X_2)\)中相应非零元素的构造取决于\({\text {ind}}(\pi _1+\pi _2)\)。此外,我们还研究了这个群对于地场的扩展的三重性(非三重性)问题。
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来源期刊
Manuscripta Mathematica
Manuscripta Mathematica 数学-数学
CiteScore
1.40
自引率
0.00%
发文量
86
审稿时长
6-12 weeks
期刊介绍: manuscripta mathematica was founded in 1969 to provide a forum for the rapid communication of advances in mathematical research. Edited by an international board whose members represent a wide spectrum of research interests, manuscripta mathematica is now recognized as a leading source of information on the latest mathematical results.
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