On the explicit Torsion Anomalous Conjecture

Abstract

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 produc 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.

An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

“…In this article we generalize the result of [CVV15] in two directions. We first present the explicit computations needed to extend the method introduced in [CVV15] to the case of E with CM.…”

“…In the last years, we have been working to approach the problem with explicit methods aiming to prove new cases of the explicit Mordell Conjecture and to eventually find all the rational points on some curves. In [CVV15] and [CVV16] joint with S. Checcoli and F. Veneziano, we give an explicit bound for the Néron-Tate height of the set of points of rank one on curves of genus at least two in E N where E is without CM. The non CM assumption is technical and we handled there the easier case where the endomorphism ring of E is Z.…”

Lattices and Rational Points

“…For a point P ∈ E, we denote byĥ(P) its Néron-Tate height as defined in [24] (which is one third of the usual Néron-Tate height used also in [18]). …”

“…A central result in our approach is Lemma 7.5 of [18], which, in turn, is a consequence of [29], Lemma 3. This is a typical application of the second Minkowski Theorem to the lattice given by the group Γ P generated by the coordinates of the point P. Similar results have been introduced by Bombieri, Masser and Zannier in [15].…”

“…In a joint paper with S. Checcoli and F. Veneziano (see [18]), we prove that the points of rank one on a weak-transverse curve of E N have bounded height and we explicitly bound their height if E is non CM. We also give a non-density result for the points of rank one on a weak-transverse variety of E N .…”