1837. Gal Binyamini Point counting for foliations over number fields
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Submission date: 2 September 2020

Abstract:

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.