On the problem of detecting linear dependence for products of abelian varieties and tori

2008/11/10 by Antonella Perucca, Perucca, Antonella
Mathematics · #14K15 (Primary) 14G25 #14L10 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:14G25 #msc:14K15 #msc:14L10

paper · pdf · doi:10.48550/arxiv.0811.1495

arxiv created 2008/11/10 · arxiv updated 2009/12/01

Abstract

Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a point in G(K) and let L be a finitely generated subgroup of G(K). Suppose that for all but finitely many primes p of K the point (R mod p) belongs to (L mod p). Does it follow that R belongs to L? We answer this question affirmatively in three cases: if L is cyclic; if L is a free left End_K G-submodule of G(K); if L has a set of generators (as a group) which is a basis of a free left End_K G-submodule of G(K). In general we prove that there exists an integer m (depending only on G, K and the rank of L) such that mR belongs to the left End_K G-submodule of G(K) generated by L.

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