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On the Explicit Torsion Anomalous Conjecture

Sara Checcoli, Francesco Veneziano, Evelina Viada

Abstract

The Torsion Anomalous Conjecture states that an irreducible variety $V$ embedded in a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a totally explicit bound for the Néron-Tate height of all maximal $V$-torsion anomalous points of relative codimension one, in the non CM case, and an analogous effective result in the CM case. As an application, we obtain the finiteness of such points. In addition, we deduce some new explicit results in the context of the effective Mordell-Lang Conjecture; in particular we bound the Néron-Tate height of the rational points of an explicit family of curves of increasing genus.

On the Explicit Torsion Anomalous Conjecture

Abstract

The Torsion Anomalous Conjecture states that an irreducible variety embedded in a semi-abelian variety contains only finitely many maximal -torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a totally explicit bound for the Néron-Tate height of all maximal -torsion anomalous points of relative codimension one, in the non CM case, and an analogous effective result in the CM case. As an application, we obtain the finiteness of such points. In addition, we deduce some new explicit results in the context of the effective Mordell-Lang Conjecture; in particular we bound the Néron-Tate height of the rational points of an explicit family of curves of increasing genus.

Paper Structure

This paper contains 22 sections, 14 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Embeddings, heights and algebraic subgroups
  3. Segre embedding
  4. Heights
  5. Algebraic Subgroups
  6. The main result
  7. An application to the Effective Mordell-Lang Conjecture
  8. Rational points on an explicit family of curves
  9. Estimates for degrees of maps
  10. Estimates for degrees of rational functions
  11. The multiplication by $m$
  12. Estimates for linear maps
  13. Estimates for the addition map
  14. Estimates for group morphisms
  15. Conclusion
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $V$ be an irreducible variety embedded in $E^N$. Then the set of maximal $V$-torsion anomalous points of relative codimension one has effectively bounded Néron-Tate height. If $E$ is non CM the bound is explicit, we have where $\omega_r=\pi^{r/2}/\Gamma(r/2+1)$ is the volume of the euclidean unit ball in $\mathbb{R}^r$, $h(V)$ is the normalised height of $V$ and $h_{\mathcal{W}}(E)$ is the h

Theorems & Definitions (27)

  • Conjecture : TAC
  • Conjecture : BHC'
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2: Arithmetic Bézout Theorem
  • Theorem 2.3: Zhang's inequality
  • ...and 17 more