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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.

We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in Checcoli and Zannier's paper [2] and obtaining relevant generalizations for them. In particular we show that that for an infinite Galois extension of the rationals the following three properties are equivalent: having uniformly bounded local degrees at every prime; having uniformly bounded local degrees at almost every prime; having Galois group of finite exponent. The proof of this result enlightens interesting connections with Zelmanov's work on the Restricted Burnside Problem. We give a formula to explicitly compute bounds for the local degrees of an infinite extension in some special cases. We relate the uniform boundedness of the local degrees to other properties: being a subfield of Q (d) , which is defined as the compositum of all number fields of degree at most d over Q; being generated by elements of bounded degree. We prove that the above properties are equivalent for abelian extensions, but not in general; we provide counterexamples based on group-theoretical constructions with extraspecial groups and their modules, for which we give explicit realizations.

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.

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.