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Rumely’s local global principle for algebraic P$$\mathcal{S}$$C fields over rings. (English) Zbl 0924.11092
Let $$S$$ be a finite set of rational primes and denote by $$N$$ the maximal Galois extension of $$\mathbb{Q}$$ in which all $$p\in S$$ decompose completely. For all elements $$\sigma_1,\dots, \sigma_e$$ of the absolute Galois group $$G(\mathbb{Q})$$ of $$\mathbb{Q}$$ let $$N(\sigma)= N(\sigma_1,\dots, \sigma_e)$$ be the fixed field of $$\langle \sigma_1,\dots, \sigma_e\rangle$$ and denote by $$\mathbb{Z}_{N(\sigma)}$$ the ring of algebraic integers in $$N(\sigma)$$. The authors prove the following result: For almost all (in the sense of the Haar measure) $$\sigma\in G(\mathbb{Q})^e$$, the field $$M= N(\sigma)$$ satisfies the following local-global-principle: Let $$V$$ be an absolutely irreducible affine variety defined over $$M$$. Suppose that for every valuation $$v$$ of $$M$$, $$V$$ admits an integral point in the henselization of $$M$$ w.r.t. $$v$$, which is simple in case $$v$$ extends some prime of $$S$$. Then $$V$$ admits a point in $$\mathbb{Z}_M^e$$.

##### MSC:
 11R58 Arithmetic theory of algebraic function fields 11R04 Algebraic numbers; rings of algebraic integers 11U05 Decidability (number-theoretic aspects) 12E25 Hilbertian fields; Hilbert’s irreducibility theorem 14G05 Rational points
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