Strong approximation and Hasse principle for integral quadratic forms over affine curves
type: Item
instance of: scholarly article
title: Strong approximation and Hasse principle for integral quadratic forms over affine curves
author: Yong Hu
author: Jing Liu
author: Yisheng Tian
published in: Acta Arithmetica
publication date: 2024-12-16T00:00:00Z
review text: In this paper, the authors start by clarifying several situations where the genus theory of integral quadratic forms (or quadratic lattices in geometric terms) over a general Dedekind domain \(R\) can be useful. In Section 2, they show that the proper spinor class number \(g^+(L)\) of a lattice \(L\) can be bounded by an idelic index associated with \(L\), and under some assumptions on local spinor norms, \(g^+(L)\) can also be bounded by the size of a 2-torsion quotient of the Picard group of \(R\). Then they prove that analogously to the global field case, proper spinor classes of \(L\) coincide with its proper classes provided that the spin group of the quadratic space \(V\) spanned by \(L\) satisfies strong approximation over \(R\) (Theorem 3.2). This result is used to deduce two sufficient conditions for the (integral) Hasse principle for representations of quadratic lattices (Theorem 3.5). It is known that the spin group \(\textbf{Spin}(V )\) satisfies strong approximation when V is isotropic. So, in that case they obtain several applications of the integral Hasse principle when \(R = k[C]\) for a curve \(C\) defined over a field \(k\) with some special properties. Finally, taking \(R\) to be a polynomial ring over a \(C_1\)-field \(k\) (i.e., every homogeneous polynomial equation of degree \(d\) over \(k\) in more than \(d\) variables has a nontrivial solution), they give an example to show that when \(V\) is anisotropic, the integral Hasse principle for the representability by \(L\) and strong approximation of \(\textbf{Spin}(V )\) can fail (Theorem 4.2). In their example the group \(\textbf{Spin}(V )\) can also be viewed as a simply connected group of inner type \(\text{A}\). So this shows that the isotropy assumption cannot be dropped in Theorem 4.3 of [\textit{Y. Hu} and \textit{Y. Tian}, ``Trivialité des groupes de Whitehead réduits avec applications à l'approximation faible et l'approximation forte'', Preprint, \url{arXiv:2401.05017}], where strong appr…
reviewed by: Adam Chapman
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 7958567
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: strong approximation
zbMATH Keywords: Hasse principle
zbMATH Keywords: spinor genus
zbMATH Keywords: spin group
MaRDI profile type: Publication
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DOI: 10.4064/AA240111-9-7
description: scientific article; zbMATH DE number 7958567