On fields of algebraic numbers with bounded local degrees

We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees. 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 the rationals; 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

On torsion anomalous intersections

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

The explicit Mordell Conjecture for families of curves (with an appendix by M. Stoll)

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.Comment: 42 pages, 1 figure, 1 tabl

On the Explicit Torsion Anomalous Conjecture

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 N\'eron-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 N\'eron-Tate height of the rational points of an explicit family of curves of increasing genus.Comment: Accepted for publication on Transactions of the American Mathematical Societ

On the Northcott property and other properties related to polynomial mappings

We prove that if K/ℚ is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/ℚ has the Northcott property. Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km /ℚ)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in ℚ(d). We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewic