Strong approximation and Hasse principle for integral quadratic forms over affine curves (Q6653257)

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 approximation is proved for isotropic groups of the same type (if \(k\) is a \(C_1\)-field).