Recent zbMATH articles in MSC 11Ehttps://www.zbmath.org/atom/cc/11E2021-02-27T13:50:00+00:00WerkzeugOn the integral representation of binary quadratic forms and the Artin condition.https://www.zbmath.org/1453.110522021-02-27T13:50:00+00:00"Deng, Yingpu"https://www.zbmath.org/authors/?q=ai:deng.yingpu"Lv, Chang"https://www.zbmath.org/authors/?q=ai:lv.chang"Shentu, Junchao"https://www.zbmath.org/authors/?q=ai:shentu.junchaoSummary: For diophantine equations of the form \(ax^2+bxy+cy^2+g=0\) over \(\mathbb{Z}\) whose coefficients satisfy some assumptions, we show that a condition with respect to the Artin reciprocity map, which we call the Artin condition, is the only obstruction to the local-global principle for integral solutions of the equation. Some concrete examples are presented.Subtle characteristic classes and Hermitian forms.https://www.zbmath.org/1453.140692021-02-27T13:50:00+00:00"Tanania, Fabio"https://www.zbmath.org/authors/?q=ai:tanania.fabioSummary: Following \textit{A. Smirnov} and \textit{A. Vishik} [``Subtle characteristic classes'', Preprint, \url{arXiv:1401.6661v1}], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfister form. Moreover, we present a description of the motive of the torsor corresponding to a hermitian form in terms of its subtle characteristic classes.Witt rings of quadratically presentable fields.https://www.zbmath.org/1453.110532021-02-27T13:50:00+00:00"Gładki, Paweł"https://www.zbmath.org/authors/?q=ai:gladki.pawel"Worytkiewicz, Krzysztof"https://www.zbmath.org/authors/?q=ai:worytkiewicz.krzysztofThe paper under discussion introduces an approach to the axiomatic theory of quadratic forms based on presentable partially ordered sets. In order to fully understand the content of this paper, one needs to follow carefully all the definitions, one-by-one, something which goes way beyond what is reasonable in a short review of the paper. In short, it can be outlined as follows: given a field \(k\), one can obtain the quadratic hyper-field \(Q(k)\). In turn, the latter gives rise to a ``presentable field'' (see definition in the paper) \(\mathcal{P}^*(Q(k))\), for which the authors define a Witt ring \(W(\mathcal{P}^*(Q(k)))\) that satisfies Witt cancellation and other nice properties that the standard Witt ring of nondegenerate symmetric bilinear forms over a given field satisfies. Among the several results appearing in this paper, the most notable one is Theorem 5.14, which states that \(W(\mathcal{P}^*(Q(k)))\) is actually the same ring as the usual Witt ring \(W(k)\) of the field \(k\).
As a whole, this paper gives an interesting alternative look on the construction of the Witt ring.
Reviewer: Adam Chapman (Tel Hai)