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Hasse principle for linear dependence in Mordell-Weil groups. (English) Zbl 1473.14042
Let $$K$$ be a number field and $$A$$ an abelian variety defined over $$K.$$ The author studies the local-global principle for linear dependence of points for abelian varieties with $${\mathrm{End}}_{\bar K}(A)=\mathbb Z.$$ The linear dependence of points $$P_1,\dots , P_n \in A(K)$$ or $$A(k_v)$$ means that $$a_{1}P_1+\dots +a_nP_n=0$$ for rational integers $$a_1,\dots , a_n$$ such that $$\gcd (a_1,\dots , a_n)$$ divides the order of the torsion subgroup of $$A(K).$$ The main result od the paper is that for a finite set of points $$S$$ the equivalence of the following statements:
1
$$S$$ is linearly dependent
2
For almost all primes $$v$$ the set of images of elements of $$S$$ via the reduction map $$r_v: A(K)\rightarrow A(k_v)$$ is linearly dependent in $$A(k_v)$$
holds iff $${\mathrm{rank}} A\leq 2\dim A.$$ The corresponding result for elliptic curves is proven without the assumption on the endomorphism ring.
##### MSC:
 14G05 Rational points 11G10 Abelian varieties of dimension $$> 1$$ 14H52 Elliptic curves
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##### References:
 [1] S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35 (2017), no. 2, 206-211. · Zbl 1391.37089 [2] S. Barańczuk, On reduction maps and support problem in $$K$$-theory and abelian varieties, J. Number Theory 119 (2006), no. 1, 1-17. · Zbl 1107.14033 [3] Y. Flicker, P. Krasoń, Multiplicative relations of points on algebraic groups, Bull. Pol. Acad. Sci. Math. 65 (2017), no. 2, 125-138. · Zbl 1409.11026 [4] J.-P. Serre, A course in Arithmetic, Graduate Texts in Mathematics, Springer 1996. [5] J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Volume 106, 2009. · Zbl 1194.11005
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