A counterexample to the local–global principle of linear dependence for Abelian varieties

Abstract

Abstract. Let A be an abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k). Gajda in 2002 asked whether it is true that the point P belongs to X if and only if the point (P mod p) belongs to (X mod p) for all but finitely many primes p of k. We answer negatively to Gajda's question.Let A be an abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k). Suppose that for all but finitely many primes p of k the point (P mod p) belongs to (X mod p). Is it true that P belongs to X? This question, which was formulated by Gajda in 2002, was named the problem of detecting linear dependence. The problem was addressed in several papers (see the bibliography) but the question was still open. In this note we show that the answer to Gajda's question is negative by providing a counterexample.Let k be a number field and let E be an elliptic curve without complex multiplication over k such that there are points P 1 , P 2 , P 3 in E(k) which are Z-linearly independent. Let P ∈ E 3 (k) and X ⊆ E 3 (k) be the following:So the group X consists of the images of the point P via the endomorphisms of E 3 with trace zero. Since the points P i are Z-independent and since E has no complex multiplication, the point P does not belong to X. Notice that no non-zero multiple of P belongs to X.Claim. Let p be a prime of k where E has good reduction. The image of P under the reduction map modulo p belongs to the image of X.For the rest of this note, we fix a prime p of good reduction for E. We write κ for the residue field of k at p. To ease notation, we now let E denote the reduction of the given elliptic curve modulo p and write P 1 , P 2 , P 3 , P and X for the image of the given points and the given subgroup under the reduction map modulo p. Our aim is to find an integer matrix M of trace zero such that P = M P in E 3 (κ).

Untitled

“…)ޑ‬ This 1-motive M is of special interest because it produces a counterexample to the so-called problem of detecting linear dependence: although P / ∈ X and even n P / ∈ X for all n = 0, there exists for every prime p where E has good reduction an element x ∈ X such that P is congruent to x modulo p. The verification of this is similar to the proof of Lemma 6.3; see [Jossen and Perucca 2010]. Using Theorem 3.1, one shows that H 1 * (l M , V M) is nontrivial -this is what makes the counterexample work and also how it was found in the first place.…”

“…By [Jossen and Perucca 2010], there exists an element x ∈ X such that red p (P) = red p (x) in A(‫ކ‬ p ). Because ‫ޚ‬ p is uniquely -divisible, we can define Q i := −i n(P − x) and get i Q i + nx = n P.…”

On the second Tate–Shafarevich group of a 1-motive

“…The map u is injective, and I will use X as a shorthand for the group u(Y ) ⊆ A(Q). This 1-motive M is of special interest because it produces a counterexample to the so called problem of detecting linear dependence: Although P / ∈ X, and even nP / ∈ X for all n = 0, there exists for every prime p where E has good reduction an element x ∈ X such that P is congruent to x modulo p. The verification of this is similar to the proof of Lemma 6.3, see [JP10]. Using Theorem 3.1 one shows that H 1 * (l M , V ℓ M ) is nontrivial -this is what makes the counterexample work, and also how it was found in the first place.…”

“…By [JP10] there exists an element x ∈ X such that red p (P ) = red p (x) in A(F p ). Because Z p is uniquely ℓ-divisible we can define Q i := ℓ −i n(P − x), and get ℓ i Q i + nx = nP .…”

Note on linear relations in Galois cohomology and {é}tale $K$-theory of curves

“…where the map ψ F,l ⊗ Z l (n) is a natural imbedding which comes from the Kummer map (5.6) ( see [BGK03] discussion on p.148 ). Our proof is a generalization of the counterexample to local -global principle for abelian varieties constructed by P. Jossen and A.Perucca in [JP10] and later extended to the context of t-modules in [BoK18]. Because of (4.8) and the choice of F we may assume that A = A e 1 1 where A 1 is a geometrically simple abelian variety.…”