Detecting linear dependence on an abelian variety via reduction maps

Abstract

Let A be a geometrically simple abelian variety over a number field k, let X be a subgroup of A(k) and let P ∈ A(k) be a rational point. We prove that if P belongs to X modulo almost all primes of k then P already belongs to X.

Untitled

“…This generalises Lemma 4.1 in [Jossen 2013a], which we get back by taking for D the trivial group. In our application, T will be T M for a 1-motive M, L will be L M , i.e., the image of := Gal(k | k) in GL(T M), and D will be the image in GL(T M) of a decomposition group D v ⊆ .…”

“…Because of the hypothesis (1), Lemma 4.4 of [Jossen 2013a] applies, which yields v ∈ T + x∈l ker(x) = T + V l and completes the proof of ( ‡).…”

“…This D is a Lie subgroup of L M , and by hypothesis, z restricts to zero in H 1 (D, T M). By Lemma 5.1 (using again [Jossen 2013a, Propositions 3.1 and 3.2]), we conclude that there is an open subgroup U of L M containing D such that z is already zero in H 1 (U, T M). This shows as well that there is an open subgroup of containing D v such that x maps to zero in H 1 ( , T M).…”

“…A positive answer to this question was given in [Jossen 2013a] in the case where G is a geometrically simple abelian variety. In this case, we know that H 1 * (k, T M) is trivial and get a nice local-global principle for subgroups of Mordell-Weil groups.…”

On a local to global principle in étale K-groups of curves

“…His theorem (cf. [17], Theorem 2, p.398) is in fact the positive answer to the analogous problem for a subgroup ƒ of the F rational points of the algebraic group G m =F: The analogous problem for abelian varieties was studied in [4], [5], [6] [10], [12], [15], [19]. For a more expanded history of the problem see [5].…”