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Preprint Number 1837

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1837. Gal Binyamini
Point counting for foliations over number fields

Submission date: 2 September 2020


We consider an algebraic variety and its foliation, both defined over a number field. We prove upper bounds for the geometric complexity of the intersection between a leaf of the foliation and a subvariety of complementary dimension (also defined over a number field). Our bounds depend polynomially on the degrees, logarithmic heights, and the logarithmic distance to a certain locus of unlikely intersections. Under suitable conditions on the foliation, we show that this implies a bound, polynomial in the degree and height, for the number of algebraic points on transcendental sets defined using such foliations.
We deduce several results in Diophantine geometry. i) Following Masser-Zannier, we prove that given a pair of sections P,Q of a non-isotrivial 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 the degrees and log-heights of the sections P,Q. In particular the set of such simultaneous torsion points is effectively computable in polynomial time. ii) Following Pila, we prove that given V ⊂ℂ^n there is an (ineffective) upper bound, polynomial in the degree and log-height of V, for the degrees and discriminants of maximal special subvarieties. In particular it follows that André-Oort for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). iii) 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.

Mathematics Subject Classification: 14G05, 11G50, 03C64, 14G35

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Full text arXiv 2009.00892: pdf, ps.

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