Analytic curves in algebraic varieties over number fields

Abstract : We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and Pólya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and p-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces, of these arithmetic criteria.
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Contributor : Antoine Chambert-Loir <>
Submitted on : Monday, February 26, 2007 - 11:38:44 AM
Last modification on : Wednesday, April 24, 2019 - 8:28:01 AM

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Jean-Benoît Bost, Antoine Chambert-Loir. Analytic curves in algebraic varieties over number fields. Yuri Tschinkel and Yuri Zarhin. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Birkhäuser, pp.69-124, 2009, Progress in mathematics n° 269. ⟨hal-00133406⟩

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