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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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