Evaluation maps, slopes, and algebraicity criteria

Abstract

We discuss criteria for the algebraicity of a formal subscheme V^ in the completion X^P at some rational point P of an algebraic variety X over some field K. In particular we consider the case where K is a function field or a number field, and we discuss applications concerning the algebraicity of leaves of algebraic foliations, algebraic groups, absolute Tate cycles, and the rationality of germs of formal functions on a curve over a number field.