论两个圆锥的乘积的还原无ramified 维特群

Pub Date : 2024-09-12 DOI:10.1007/s00229-024-01591-x
Alexander S. Sivatski
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引用次数: 0

摘要

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。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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

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

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.

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