On counterexamples to local-global divisibility in commutative algebraic groups

Abstract

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.

On the local-global divisibility over abelian varieties

“…They also found a counterexample to the local-global divisibility by powers of 3 for an elliptic curve over Q. From this result, the examples of Dvornicich and Zannier [DZ2] and Paladino [Pal], [Pal2] and the main result of [PRV2], it follows that the set of prime numbers q for which there exists an elliptic curve E ′ defined over Q and n ∈ N such that the local-global divisibility by q n does not hold over E ′ (Q) is just {2, 3}.…”

Weil–Châtelet divisible elements in Tate–Shafarevich groups II: On a question of Cassels

“…Cases where {a ∈ A(k); a ∈ p n · A(k v ) for all v } = p n · A(k) are known, even when k = Q and A = E is an elliptic curve: see [DZ04] for p n = 4, and more generally in [Pa10]. These examples yield in particular cases where X 1 (k, A p n ) = 0, but in view of Proposition 14 and (4.4) this does not imply that Cassels' question has a negative answer for A t /k.…”

On the minimal set for counterexamples to the local–global principle

“…However, it remains an open question whether 3 belongs or not to S Q . Unfortunately, for powers of 3, all known counterexamples are given for some elliptic curves defined over non-trivial extensions of Q, see [9].…”