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.

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.Una importante congettura sulle varietà torsione-anomale afferma che se Vè una varietà debolmente-trasversa in una varietà abeliana, allora il complementare V ta di tutte le varietà V -torsione-anomaleè aperto e denso in V . In questo articolo dimostriamo alcuni casi della congettura. In particolare, mostriamo che le varietà V -torsione-anomale di codimensione relativa uno non sono dense in ogni varietà V debolmente trasversa, immersa in un prodotto di curve ellittiche con CM. Inoltre diamo stime esplicite e uniformi nella dipendenza da V . Come immediata conseguenza otteniamo la suddetta congettura per V di codimensione due in un prodotto di curve ellittiche CM. Infine, evidenziamo alcune implicazioni sulla Congettura di Mordell-Lang Effettiva.

The theory of continued fractions has been generalized to ℓ-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic irrationals for the ℓ-adic continued fractions introduced by Ruban. In this case, rational numbers may have a periodic non-terminating continued fraction expansion; moreover, for quadratic irrational numbers, no analogue of Lagrange's theorem holds. We give general explicit criteria to establish the periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol [Lao85]).

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin-Demjanenko and the analysis of our explicit examples is carried to conclusion.

This chapter details Umberto Zannier's minicourse on hyperelliptic continued fractions and generalized Jacobians. It begins by presenting the Pell equation, which was studied by Indian, and later by Arabic and Greek, mathematicians. The chapter then addresses two questions about continued fractions of algebraic functions. The first concerns the behavior of the solvability of the polynomial Pell equation for families of polynomials. It must be noted that these questions are related to problems of unlikely intersections in families of Jacobians of hyperelliptic curves (or generalized Jacobians). The chapter also reviews several classical definitions and results related to the continued fraction expansion of real numbers and illustrates them by examples.