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- W2059063923 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 $mathfrak{p}incal{S}$ 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 $mathfrak{q}incal{V}_M$ let $hat{cal{O}}_{M,mathfrak{q}}={xinhat{M}_{mathfrak{q}}|;|x|_{mathfrak{q}}le 1}$, where $hat{M}_{mathfrak{q}}$ is a completion of $M$ at $mathfrak{q}$, and let $cal{O}_{M,cal{V}}={xin M|;|x|_{mathfrak{q}}le 1 hbox{{rm for each}} mathfrak{q}incal{V}_M}$. For $bf{sigma}=(sigma_1,dots,sigma_e)in Gal(K)^e$, let $K_s(bf{sigma})={xin K_s |; sigma_i(x)=x,,i=1,dots,e}$. 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 $Vsubseteqmathbb{A}^n$ 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 $mathfrak{q}incal{V}_Msetminuscal{S}_M$ and $bf{x}_{mathfrak{q}}in V_{rm simp}(hat{cal{O}}_{M,mathfrak{q}})$ for each $mathfrak{q}incal{S}_M$ such that $|x_{i,mathfrak{q}}|_{mathfrak{q}} lt 1$, $i=1,dots,n$, for each archimedean prime $mathfrak{q}incal{V}_M$. Then $V(cal{O}_{M,cal{V}})neemptyset$." @default.
- W2059063923 created "2016-06-24" @default.
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- W2059063923 date "2008-11-01" @default.
- W2059063923 modified "2023-09-26" @default.
- W2059063923 title "Rumely's local global principle for Weakly P$calS$C Fields over Holomorphy Domains" @default.
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- W2059063923 doi "https://doi.org/10.7169/facm/1229696552" @default.
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