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 BorelDwork and PólyaBertrandias 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2007
 arXiv:
 arXiv:math/0702593
 Bibcode:
 2007math......2593B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 14G40;
 31A15;
 14G22
 EPrint:
 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin", Y. Tschinkel &