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Algebra, Arithmetic, and Geometry pp 69–124Cite as

Analytic Curves in Algebraic Varieties over Number Fields

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Part of the book series: Progress in Mathematics ((PM,volume 269))

Summary

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 on algebraic curves that extends the classical rationality theorems of Borel–Dwork and Polya–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.

2000 Mathematics Subject Classifications: 14G40 (Primary); 14G22, 31A15, 14B20 (Secondary)

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Notes

  1. 1.

    Since the first version of this paper was written, the relevance of rigid analytic geometry à la Berkovich to develop a non-archimedean potential theory on p-adic curves, and consequently a “modern” version of Arakelov geometry of arithmetic surfaces satisfying the above principle of “equality of places,” has been largely demonstrated by A. Thuillier in his thesis [51].

  2. 2.

    Our terminology differs slightly from that in [11]. In the present article, the term capacitary metric will be used for two distinct notions: for the metrics on line bundles defined using equilibrium potentials just defined, and for some metrics on the tangent line to M at a point; see Section 5.C. In [11], it was used for the latter notion only.

  3. 3.

    The proofs in both references are similar and rely on the Abel–Jacobi map, together with the fact that K is the union of its locally compact subfields.

  4. 4.

    In the terminology of [11], nonnegative.

  5. 5.

    Using the fact that bounded subsets of \({\bf C}_p\) are contained in affinoids (actually, lemniscates) with arbitrarily close transfinite diameters.

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Authors and Affiliations

  1. Département de mathématiques, Université de Paris-Sud (Orsay), Bâtiment 425, F-91405, Orsay Cedex, France

    Jean-Benoît Bost

  2. Université de Rennes 1, Institut de recherche mathématique de Rennes, Campus de Beaulieu, F-35042, Rennes Cedex, France

    Antoine Chambert-Loir

Authors
  1. Jean-Benoît Bost
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Correspondence to Jean-Benoît Bost .

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Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

  2. Eberly College of Science, Dept. Mathematics, Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Yuri Zarhin

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Bost, JB., Chambert-Loir, A. (2009). Analytic Curves in Algebraic Varieties over Number Fields. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_3

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