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

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

2008-11-10

arxiv.org/abs/0811.1495v1

Mathematics

Number Theory

Scientific paper

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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