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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:
\(S\) is linearly dependent
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.
14G05 Rational points
11G10 Abelian varieties of dimension \(> 1\)
14H52 Elliptic curves
Full Text: DOI Euclid
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