Rumely's local global principle for weakly P\(\mathcal S\)C fields over holomorphy domains
Let \(K\) be a global field, \(\mathcal{V}\) an infinite proper subset of the set of all primes of \(K\), and \(\mathcal{S}\) a finite subset of \(\mathcal{V}\). Denote the maximal Galois extension of \(K\) in which each \(\mathfrak{p}\in\mathcal{S}\) totally splits by \(\mathbb K_{tot,S}\). Let \(M\) be an algebraic extension of \(K\). Let \(\mathcal{V}_M\) (resp. \(\mathcal{S}_M\)) be the set of primes of \(M\) which lie over primes in \(\mathcal{V}\) (resp. \(\mathcal{S}\)). For each \(\mathfrak{q}\in\mathcal{V}_M\) let \(\hat{\mathcal{O}}_{M,\mathfrak{q}}=\{x\in\hat{M}_{\mathfrak{q}}|\;|x|_{\mathfrak{q}}\leq 1\}\), where \(\hat{M}_{\mathfrak{q}}\) is a completion of \(M\) at \(\mathfrak{q}\), and let \(\mathcal{O}_{M,\mathcal{V}}=\{x\in M\mid |x|_{\mathfrak{q}}\leq 1\;\hbox{{\mathrm for each}} \mathfrak{q}\in\mathcal{V}_M\}\). For \(\mathbf{sig}=(\text{sig}_1,\dots,\text{sig}_e)\in\text{Gal}(K)^e\), let \(K_s(\mathbf{sig})=\{x\in K_s \mid \text{sig}_i(x)=x,\,i=1,\dots,e\}\). Then, for almost all \(\mathbf{sig}\in\text{Gal}(K)^e\) (with respect to the Haar measure), the field \(M=K_s(\mathbf{sig})\cap\mathbb K_{tot,S}\) satisfies the following local global principle: Let \(V\subseteq\mathbb{A}^n\) be an affine absolutely irreducible variety defined over \(M\). Suppose that there exist \(\mathbf{x}_{\mathfrak{q}}\in V(\hat{\mathcal{O}}_{M,\mathfrak{q}})\) for each \(\mathfrak{q}\in\mathcal{V}_M\setminus\mathcal{S}_M\) and \(\mathbf{x}_{\mathfrak{q}}\in V_{\text{simp}}(\hat{\mathcal{O}}_{M,\mathfrak{q}})\) for each \(\mathfrak{q}\in\mathcal{S}_M\) such that \(|x_{i,\mathfrak{q}}|_{\mathfrak{q}}<1\), \(i=1,\dots,n\), for each archimedean prime \(\mathfrak{q}\in\mathcal{V}_M\). Then \(V(\mathcal{O}_{M,\mathcal{V}})\neq\emptyset\).
scientific article; zbMATH DE number 5635535
scientific article