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On torsion anomalous intersections

Sara Checcoli, Francesco Veneziano, Evelina Viada

Abstract

A deep conjecture on torsion anomalous varieties states that if $V$ is a weak-transverse variety in an abelian variety, then the complement $V^{ta}$ of all $V$-torsion anomalous varieties is open and dense in $V$. We prove some cases of this conjecture. We show that the $V$-torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety $V$ embedded in a product of elliptic curves with CM. We give explicit uniform bounds in the dependence on $V$. As an immediate consequence we prove the conjecture for $V$ of codimension two in a product of CM elliptic curves. We also point out some implications on the effective Mordell-Lang Conjecture.

On torsion anomalous intersections

Abstract

A deep conjecture on torsion anomalous varieties states that if is a weak-transverse variety in an abelian variety, then the complement of all -torsion anomalous varieties is open and dense in . We prove some cases of this conjecture. We show that the -torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety embedded in a product of elliptic curves with CM. We give explicit uniform bounds in the dependence on . As an immediate consequence we prove the conjecture for of codimension two in a product of CM elliptic curves. We also point out some implications on the effective Mordell-Lang Conjecture.

Paper Structure

This paper contains 19 sections, 23 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Applications to the Mordell-Lang Conjecture
  3. Torsion anomalous varieties: preliminary results
  4. Maximality and minimality
  5. Conjecture \ref{['TORAN']} implies the Torsion Openness Conjecture and the Torsion Finiteness Conjecture
  6. Relative position of the torsion anomalous varieties
  7. Torsion anomalous varieties as components of different intersections
  8. No torsion anomalous varieties on a weak-transverse translate
  9. Finitely many maximal $V$-torsion anomalous varieties in $B+\mathrm{Tor}_G$
  10. Main ingredients
  11. Notation
  12. Subgroups and torsion varieties
  13. The Zhang Estimate
  14. The Arithmetic Bézout theorem
  15. An effective Bogomolov Estimate for relative transverse varieties
  16. ...and 4 more sections

Key Result

Theorem 1.4

Let $V$ be a weak-transverse variety in a product $E^N$ of elliptic curves with CM defined over a number field $k$. Then the maximal $V$-torsion anomalous varieties $Y$ of relative codimension one are finitely many. In addition the degrees and normalized heights are bounded as follows. For any posit

Theorems & Definitions (47)

  • Conjecture 1.1: CIT, Conjecture on the Intersection with Torsion Varieties
  • Definition 1.2
  • Conjecture 1.3: Bombieri-Masser-Zannier
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • proof
  • Corollary 1.7
  • Corollary 2.1
  • ...and 37 more