Point counting for foliations over number fields

LetM be an affine variety equipped with a foliation, both defined over a number field K. For an algebraic V⊂M over K, write δV for the maximum of the degree and log-height of V. Write ΣV for the points where the leaves intersect V improperly. Fix a compact subset B of a leaf L. We prove effective bounds on the geometry of the intersection B∩V. In particular, when codimV=dimL we prove that #(B∩V) is bounded by a polynomial in δV and logdist−1(B,ΣV). Using these bounds we prove a result on the interpolation of algebraic points in images of B∩V by an algebraic map Φ. For instance, under suitable conditions we show that Φ(B∩V) contains at most poly(g,h) algebraic points of log-height h and degree g.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 δP,δQ; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given V⊂Cn, there is an (ineffective) upper bound, polynomial in δV, 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.