Local–global divisibility by 4 in elliptic curves defined over $${\mathbb{Q}}$$

Abstract

Let E be an elliptic curve defined over Q. Let P ∈ E(Q) and let q be a positive integer. Assume that for almost all valuations v ∈ Q, there exist pointsIs it possible to conclude that there exists a point D ∈ E(Q) such that P = q D? A full answer to this question is known when q is a power of almost all primes p ∈ N, but some cases remain open when p ∈ S = {2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67, 163}. We now give a complete answer in the case when q = 4.

Divisibility questions in commutative algebraic groups

“…In fact, for p ≤ n/2+1, the p-Sylow subgroup of G could be a direct product of two cyclic groups C p (look for examples at the 3-Sylow subgroup of of A 7 , when n = 4, or at the 3-Sylow subgroup M 11 when n = 5, and so on). When the p-Sylow subgroup G p of G is isomorphic to C 2 p , the local-global divisibility may fail as in the mentioned examples produced in [14], [16] and in [29], [30], [31]. In principle one could rise those examples to similar ones for all n. So the local-global principle for divisibility by p could fail for p ≤ n/2 + 1, when G p ≃ C 2 p .…”

“…Moreover we use the description of subgroups of GL n (q) of class C 9 given in Aschbacher's Theorem and recalled in Section 2. When the p-Sylow subgroup of G is isomorphic to C 2 p , there are known counterexample to the local-global divisibility (see the mentioned [14], [15], [30]) and even when the p-Sylow subgroup of G is isomorphic to C 3 p there are counterexamples (see [29]). In various parts of the proof we will show that for p > n/2 + 1, the p-Sylow subgroup of G is either trivial or cyclic (which does not hold for p ≤ n/2 + 1).…”

Counterexamples to the local–global divisibility over elliptic curves

“…Many mathematicians got criterions for the validity of the local-global divisibility principle for many families of commutative algebraic groups, as algebraic tori ( [DZ1] and [Ill]), elliptic curves ( [Cre1], [Cre2], [DZ1], [DZ2], [DZ3], [GR1], [Pal1], [Pal2], [PRV1], [PRV2]), and very recently polarized abelian surfaces ( [GR2]) and GL 2 -type varieties ( [GR3]).…”

Fields generated by torsion points of elliptic curves

“…This kind of information is useful for describing the fields in terms of degrees and Galois groups, as we shall explicitly show for m = 3 and m = 4, when char(K) = 2, 3. Other applications are local-global problems (see, e.g., [5] or the particular cases of [10] and [11]), descent problems (see, e.g., [13] and the references there or, for a particular case, [2] and [3]), Galois representations, points on modular curves (see Section 4) and points on Shimura curves.…”