Analytic curves in algebraic varieties over number fields

Details Analytic curves in algebraic varieties over number fields Analytic curves in algebraic varieties over number fields

2007-02-20

arxiv.org/abs/math/0702593v3

Mathematics

Number Theory

55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"a

Scientific paper

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 P\'olya-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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