Descent obstructions and Brauer-Manin obstruction in positive characteristic (Q2841760)

The authors prove that the Brauer-Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global function fields.NEWLINENEWLINELet \(K\) be a global function field of characteristic \(p > 0\) with completions \(K_v\), \(S\) a non-empty set of primes of \(K\) and \(\mathcal{O}_S \subset K\) the ring of \(S\)-integers.NEWLINENEWLINE``\textbf{Definition 1.1.} Let \(G\) be a \(K\)-group scheme. Let \(Y \to X\) be an \(X\)-torsor under \(G\). We say that a point \((x_v) \in \prod_vX(K_v)\) is \textit{unobstructed} by \(Y\) if the evaluation \([Y]((x_v)) \in \prod_vH^1(K_v,G)\) comes from a global element \(a \in H^1(K,G)\) through the diagonal map.''NEWLINENEWLINE``\textbf{Definition 1.2.} Let \(X\) be a \(K\)-variety. An \textit{Artin-Schreier torsor} over \(X\) is a torsor under the étale group scheme \(\mathbf{F}_s\) (for some \(s = q^e\), \(e > 0\)) given by the equation \(z^s - z = g\) for some \(g \in K[T]\).''NEWLINENEWLINE``\textbf{Theorem 2.2.} Let \(\mathcal{X}\) be an affine \(\mathcal{O}_S\)-scheme of finite type with generic fibre \(X\). Let \((x_v) \in \prod_{v\not\in S}\mathcal{X}(\mathcal{O}_v) \times \prod_{v\in S}X(K_v)\). Assume that \((x_v)\) is unobstructed by every Artin-Schreier torsor \(Y \to X\). Then \((x_v) \in \mathcal{X}(\mathcal{O}_S)\).''NEWLINENEWLINE``\textbf{Corollary 3.2.} Let \(X\) be an affine variety over \(K\). Assume that \((a_v)_v \in \prod_vX(K_v)\) is unobstructed by every \(\alpha_p\)-torsor. Then \((a_v) \in X(K)\).''NEWLINENEWLINE``\textbf{Definition 4.1.} Let \(X\) be a \(K\)-variety. Set \(\mathrm{Br}_0(X) := \mathrm{Im}(\mathrm{Br}(K) \to \mathrm{Br}(X))\). For each \(e > 0\), we define a subgroup \(B_{AS,e}(X)\) of the Brauer group \(\mathrm{Br}(X)\) as the subgroup generated by \(\mathrm{Br}_0(X)\) and the cup-products \((a \cup [Y])\), where \(Y\) runs over all Artin-Schreier \(X\)-torsors under \(\mathbf{F}_{q^e}\) and \(a\) runs over all elements of \(H^1(K,G_{q^e})\). Then we set \(B_{AS}(X) := \bigcup_{e > 0}B_{AS,e}(X)\).''NEWLINENEWLINE``\textbf{Theorem 4.2.} Let \(X\) be an affine \(K\)-variety. Let \((a_v) \in \prod_vX(K_v)\) be an adelic point on \(X\). Assume that for every element \(\theta \in B_{AS}(X)\), the evaluation \(\theta((a_v))\) is global, i.e., comes from an element of \(\mathrm{Br}(K)\) by the diagonal embedding. Then \((a_v) \in X(K)\).''NEWLINENEWLINEIn Section 5, the authors construct an example of an affine scheme \(\mathcal{X}\) over \(\mathbb{A}^1_{\mathbb{F}_p}\) (so \(S\) is the point of \(\mathbb{P}^1\) at infinity) such that \(\mathcal{X}\) has points over every completion \(\mathcal{O}_S\), but \(\mathcal{X}(\mathcal{O}_S) = \emptyset\), hence a counterexample to the integral Hasse principle in characteristic \(p\).