Point counting for foliations over number fields

Abstract

Let${\mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${\mathbb K}$. For an algebraic $V\subset {\mathbb M}$ over ${\mathbb K}$, write $\delta _{V}$ for the maximum of the degree and log-height of V. Write $\Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${\mathcal B}$ of a leaf ${\mathcal L}$. We prove effective bounds on the geometry of the intersection ${\mathcal B}\cap V$. In particular, when $\operatorname {codim} V=\dim {\mathcal L}$ we prove that $\#({\mathcal B}\cap V)$ is bounded by a polynomial in $\delta _{V}$ and $\log \operatorname {dist}^{-1}({\mathcal B},\Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of ${\mathcal B}\cap V$ by an algebraic map $\Phi $. For instance, under suitable conditions we show that $\Phi ({\mathcal B}\cap V)$ contains at most $\operatorname {poly}(g,h)$ algebraic points of log-height h and degree g.

Keywords

• 14G05
• 11G50
• 03C64
• 14G35