Rumely’s local global principle for algebraic ${\mathrm P}{\mathcal S}{\mathrm C}$ fields over rings

Abstract:

Let $\mathcal {S}$ be a finite set of rational primes. We denote the maximal Galois extension of $\mathbb {Q}$ in which all $p\in \mathcal {S}$ totally decompose by $N$. We also denote the fixed field in $N$ of $e$ elements $\sigma _{1},\ldots , \sigma _{e}$ in the absolute Galois group $G( \mathbb {Q})$ of $\mathbb {Q}$ by $N( {\boldsymbol \sigma })$. We denote the ring of integers of a given algebraic extension $M$ of $\mathbb {Q}$ by $\mathbb {Z}_{M}$. We also denote the set of all valuations of $M$ (resp., which lie over $S$) by $\mathcal {V}_{M}$ (resp., $\mathcal {S}_{M}$). If $v\in \mathcal {V}_{M}$, then $O_{M,v}$ denotes the ring of integers of a Henselization of $M$ with respect to $v$. We prove that for almost all ${\boldsymbol \sigma }\in G( \mathbb {Q})^{e}$, the field $M=N( {\boldsymbol \sigma })$ satisfies the following local global principle: Let $V$ be an affine absolutely irreducible variety defined over $M$. Suppose that $V(O_{M,v})\not =\varnothing$ for each $v\in \mathcal {V}_{M}\backslash \mathcal {S}_{M}$ and $V_{\mathrm {sim}}(O_{M,v})\not =\varnothing$ for each $v\in \mathcal {S}_{M}$. Then $V(O_{M})\not =\varnothing$. We also prove two approximation theorems for $M$.