Point counting for foliations over number fields

Author:

Binyamini GalORCID

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. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Wilkie's conjecture for Pfaffian structures;Annals of Mathematics;2024-03-01

2. Rolle Models in the Real and Complex World;Handbook of Geometry and Topology of Singularities V: Foliations;2024

3. Finiteness theorems on elliptical billiards and a variant of the dynamical Mordell–Lang conjecture;Proceedings of the London Mathematical Society;2023-10-04

4. Torsion points on isogenous abelian varieties;Compositio Mathematica;2022-05

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3