[Paper Review] Point counting for foliations over number fields

Gal Binyamini|arXiv (Cornell University)|Sep 2, 2020

Algebraic Geometry and Number Theory42 references4 citations

TL;DR

This paper establishes effective polynomial bounds on the number of intersections between leaves of a foliation on a variety defined over a number field and algebraic subvarieties, using growth estimates for solutions of Fuchsian differential equations. The key result is a point counting bound polynomial in degree and log-distance to singularities, enabling effective decidability and interpolation results in Diophantine geometry, including effective bounds on simultaneous torsion points in elliptic curves and polynomial-time decidability of André-Oort for powers of modular curves.

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 \emph{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 $V\subset\mathbb{C}^n$ 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.

Motivation & Objective

• To establish effective, polynomial bounds on the number of intersections between leaves of a foliation and algebraic subvarieties over number fields.
• To develop a framework for counting algebraic points in images of such intersections via algebraic maps, with bounds in terms of degree and height.
• To apply these bounds to resolve effective versions of classical problems in Diophantine geometry, including Masser-Zannier and Pila's André-Oort conjecture.
• To derive Galois-orbit lower bounds for torsion points on elliptic curves using analytic methods, avoiding transcendence theory.

Proposed method

• Use of Fuchsian differential equations over number fields to model the geometry of foliation leaves and control solution growth.
• Application of Weierstrass polydisc covering and jet norm estimates to control local geometry of intersections.
• Establishment of effective bounds on the growth of jet norms of solutions to inhomogeneous Fuchsian equations via degree and height control.
• Construction of chains of discs with controlled radii to propagate bounds from a base point to any target point, ensuring polynomial dependence on distance to singularities.
• Use of height and degree invariants (δ_V) to quantify complexity of algebraic subvarieties and their intersections.
• Leverage of the Pila-Wilkie-type counting principle in the context of foliations to derive bounds on algebraic points in images of intersections.

Experimental results

Research questions

• RQ1 Can the number of intersections between a leaf of a foliation and an algebraic subvariety be bounded effectively in terms of the subvariety’s degree and height?
• RQ2 Is there a polynomial bound on the number of algebraic points of bounded degree and height in the image of such an intersection under an algebraic map?
• RQ3 Can the Masser-Zannier conjecture on simultaneous torsion points in families of abelian varieties be made effective with polynomial bounds in the invariants of the sections?
• RQ4 Does the André-Oort conjecture for powers of the modular curve admit a polynomial-time decidable algorithm, even if the constant is ineffective?
• RQ5 Can Galois-orbit lower bounds for torsion points on elliptic curves be derived using analytic methods rather than transcendence theory?

Key findings

• The number of intersections between a compact leaf segment and an algebraic subvariety V is bounded by a polynomial in δ_V and log(dist⁻¹(B, Σ_V)), where δ_V combines degree and log-height of V.
• For a map Φ, the number of algebraic points of degree g and log-height h in Φ(B ∩ V) is bounded by a polynomial in g and h, under suitable conditions.
• The set of simultaneous torsion points in a non-isotrivial family of squares of elliptic curves is effectively computable in polynomial time, with torsion order bounded by a polynomial in δ_P and δ_Q.
• An ineffective but polynomial upper bound is established for the degrees and discriminants of maximal special subvarieties in subvarieties of C^n, implying polynomial-time decidability of André-Oort for powers of the modular curve.
• A Galois-orbit lower bound for torsion points on elliptic curves is derived using the analytic method, matching bounds previously obtained via transcendence techniques.
• The growth of solutions to inhomogeneous Fuchsian equations is shown to be polynomial in the inverse distance to singularities, with effective dependence on the equation’s degree and height.