No abstract

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 k be an algebraic closed field of characteristic zero. Let K be the rational function field K = k(t). Let φ be a non isotrivial rational function in K(z). We prove a bound for the cardinality of the set of K-rational preperiodic points for φ in terms of the number of places of bad reduction and the degree d of φ.

On counterexamples to local-global divisibility in commutative algebraic groups by Laura Paladino (Pisa)1. Introduction. Let k be a number field and let A be a commutative algebraic group defined over k. Let P ∈ A(k). We denote by M k the set of the valuations v ∈ k and by k v the completion of k at the valuation v. In previous papers we were concerned with the following question:Problem. Assume that for all but finitely many v ∈ M k , there exists D v ∈ A(k v ) such that P = qD v , where q is a positive integer. Is it possible to conclude that there exists D ∈ A(k) such that P = qD?This problem is known as Local-Global Divisibility Problem. There are known solutions in many cases, but many cases remain open too. By using the Bézout identity, it turns out that it is sufficient to solve it in the case when q is a power p n of a prime p, to get answers for a general integer q.The local-global divisibility problem is motivated by a strong form of the Hasse principle that says: If a quadratic form ax 2 + bxy + cy 2 ∈ Q[x, y] of rank 2 represents 0 non-trivially over all but finitely many completions Q p , then it represents 0 non-trivially over Q. Then, in particular, if a rational number is a perfect square modulo all but finitely many primes p, then it is a rational square. A generalization of this fact for q-powers of k-rational numbers is the Local-Global Divisibility Problem in the case when A is the multiplicative group G m . For this algebraic group a solution is classical. The answer is affirmative for all odd prime powers q and for q | 4 (see [AT, Chap. IX, Thm. I]). On the other hand, there are counterexamples for q = 2 t , t ≥ 3. The most famous of them was discovered by Trost (see [Tro]) and it is the diophantine equation x 8 = 16, that has a solution in Q p for all primes p ∈ Q different from 2, but has no solutions in Q 2 and in Q.

Let K be a field and let E be an elliptic curve defined over K. Let m be a positive integer, prime with char(K) if char(K) = 0; we denote by E[m] the m-torsion subgroup of E and by K m := K(E[m]) the field obtained by adding to K the coordinates of the pointsWe look for small sets of generators for K m inside {x 1 , y 1 , x 2 , y 2 , ζ m } trying to emphasize the role of ζ m (a primitive m-th root of unity). In particular, we prove that K m = K(x 1 , ζ m , y 2 ), for any odd m 5. When m is prime and K is a number field we prove that the generating set {x 1 , ζ m , y 2 } is often minimal. We also describe explicit generators, degree and Galois groups of the extensions K m /K for m = 3 and m = 4, when char(K) = 2, 3. * L.