Rumely's local global principle for Weakly P$\calS$C Fields over Holomorphy Domains

Abstract

Let $K$ be a global field, $\cal{V}$ an infinite proper subset of the set of all primes of $K$, and $\cal{S}$ a finite subset of $\cal{V}$. Denote the maximal Galois extension of $K$ in which each $\mathfrak{p}\in\cal{S}$ totally splits by $K_{tot,S}$. Let $M$ be an algebraic extension of $K$. Let $\cal{V}_M$ (resp. $\cal{S}_M$) be the set of primes of $M$ which lie over primes in $\cal{V}$ (resp. $\cal{S}$). For each $\mathfrak{q}\in\cal{V}_M$ let $\hat{cal{O}}_{M,\mathfrak{q}}=\{x\in\hat{M}_{\mathfrak{q}}|\;|x|_{\mathfrak{q}}\le 1\}$, where $\hat{M}_{\mathfrak{q}}$ is a completion of $M$ at $\mathfrak{q}$, and let $\cal{O}_{M,\cal{V}}=\{x\in M|\;|x|_{\mathfrak{q}}\le 1\ \hbox{{\rm for each}} \mathfrak{q}\in\cal{V}_M\}$.

For $\bf{\sigma}=(\sigma_1,\dots,\sigma_e)\in Gal(K)^e$, let $K_s(\bf{\sigma})=\{x\in K_s |\; \sigma_i(x)=x,\,i=1,\dots,e\}$. Then, for almost all $\bf{\sigma}\in Gal(K)^e$ (with respect to the Haar measure), the field $M=K_s(\bf{\sigma})\cap K_{tot,S}$ 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 $\bf{x}_{\mathfrak{q}}\in V(\hat{\cal{O}}_{M,\mathfrak{q}})$ for each $\mathfrak{q}\in\cal{V}_M\setminus\cal{S}_M$ and $\bf{x}_{\mathfrak{q}}\in V_{\rm simp}(\hat{\cal{O}}_{M,\mathfrak{q}})$ for each $\mathfrak{q}\in\cal{S}_M$ such that $|x_{i,\mathfrak{q}}|_{\mathfrak{q}} \lt 1$, $i=1,\dots,n$, for each archimedean prime $\mathfrak{q}\in\cal{V}_M$. Then $V(\cal{O}_{M,\cal{V}})\ne\emptyset$.