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.

55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkhäuser, 2009

Cited by in corpus (6)

• Algebraization, transcendence, and D-group schemes
• Algebraic points on meromorphic curves
• Algebraic solutions of differential equations over the projective line minus three points
• Simply connected varieties in characteristic $p>0$
• Differential forms on arithmetic jet spaces
• Numerical equivalence of $\mathbb{R}$-divisors and Shioda-Tate formula for arithmetic varieties