Abstract
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.
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.
Publisher
Cambridge University Press (CUP)
Subject
Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Analysis
Cited by
4 articles.
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