Point counting for foliations over number fields

Abstract

AbstractLet${\mathbb M}$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 ofV. Write$\Sigma _{V}$for the points where the leaves intersectVimproperly. 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)$algebraic points of log-heighthand degreeg.We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections$P,Q$of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever$P,Q$are simultaneously torsion their order of torsion is bounded effectively by a polynomial in$\delta _{P},\delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given$V\subset {\mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in$\delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.