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Applications of \(p\)-adic analysis for bounding periods of subvarieties under étale maps. (English) Zbl 1349.14002

Summary: Using methods of \(p\)-adic analysis, we obtain effective bounds for the length of the orbit of a preperiodic subvariety \(Y \subset X\) under the action of an étale endomorphism of \(X\). As a corollary of our result, we obtain effective bounds for the size of torsion of any semiabelian variety over a finitely generated field of characteristic 0. Our method allows us to show that any finitely generated torsion subgroup of \(\mathrm{Aut}(X)\) is finite. This yields a different proof of Burnside’s problem for automorphisms of quasiprojective varieties \(X\) defined over a field of characteristic 0.

MSC:

14A10 Varieties and morphisms
14G20 Local ground fields in algebraic geometry
26E30 Non-Archimedean analysis
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