Let p > 3 be a prime number and let n be a positive integer. We prove that the local-global principle for divisibility by p n holds for elliptic curves defined over the rationals. For this, we refine our previous criterion for the validity of the principle. We also give an example that shows that the assumptions of our criterion are necessary.

Let p ≥ 2 be a prime number and let k be a number field. Let A be an abelian variety defined over k. We prove that if Gal(k(A[p])/k) contains an element g of order dividing p − 1 not fixing any non-is trivial, then the local-global divisibility by p n holds for A(k) for every n ∈ N. Moreover, we prove a similar result without the hypothesis on the triviality of H 1 (Gal(k(A[p])/k), A[p]), in the particular case where A is a principally polarized abelian variety. Then, we get a more precise result in the case when A has dimension 2. Finally we show with a counterexample that the hypothesis over the order of g is necessary.In the Appendix, we explain how our results are related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperani and Stix [C-S].

Let p be a prime number and let k be a number field. Let E be an elliptic curve defined over k. We prove that if p is odd, then the local-global divisibility by any power of p holds for the torsion points of E. We also show with an example that the hypothesis over p is necessary.We get a weak generalization of the result on elliptic curves to the larger family of GL 2 -type varieties over k. In the special case of the abelian surfaces A/k with quaternionic multiplication over k we obtain that for all prime p, except a finite number depending on A, the local-global divisibility by any power of p holds for the torsion points of A.The converse of Proposition 1 is not true. However, in the case when the group H 1 loc (Gal(k(A[p n ])/k), A[p n ]) is not trivial, we can find an extension L of k such that L and k(A[p n ]) are linearly disjoint over k, and such that the local-global divisibility by p n over A(L) does not hold (see [DZ3, Theorem 3] for the details).An abelian variety defined over a number field k is said to be of GL 2 -type if there is an embedding φ : E ֒→ End k (A) ⊗ Q, where E is a number field such that [E : Q] = dim(A) (see [R, Chapter 2, Chapter 5]). Let O E be the ring of the algebraic integers of E.is an order of O E which is isomorphic to some subring of End k (A) via φ. Following [R, Chapter 2], we say that a prime number is good for A if it does not divide the index [O E : R] (the notion of good depends also on E). We prove the following result, generalizing the result on elliptic curves.

We construct an infinite family of totally real algebraic extensions of [Formula: see text] whose ring of integers has a Julia Robinson number distinct from [Formula: see text] and [Formula: see text]. In fact, the set of Julia Robinson numbers obtained is unbounded. This gives new examples of algebraic extensions of [Formula: see text] of infinite degree whose ring of integers has undecidable first-order theory.

Let p ≥ 5 be a prime number. We find all the possible subgroups G of GL 2 (Z/pZ) such that there exists a number field k and an elliptic curve E defined over k such that the Gal(k(is isomorphic to the G-module (Z/pZ) 2 and there exists n ∈ N such that the local-global divisibility by p n does not hold over E(k).