Kummer theory of abelian varieties and reductions of Mordell-Weil groups

Abstract

Let A be an abelian variety over a number field F with End(A/F) commutative. Let S be a subgroup of A(F) and let x be a point of A(F). Suppose that for almost all places v of F the reduction of x modulo v lies in the reduction of S modulo v. In this paper we prove that x must then lie in S + A(F)_tors. This provides a partial answer to a generalization (due to W. Gajda) of the support problem of Erdos.

References in corpus (1)

• A support problem for the intermediate Jacobians of l-adic representations

Cited by in corpus (5)

• On a Hasse principle for Mordell-Weil groups
• Detecting linear dependence on a simple abelian variety
• On the problem of detecting linear dependence for products of abelian varieties and tori
• On a generalization of the support problem of Erdos and its analogues for abelian varieties and K-theory
• Linear dependence in Mordell-Weil groups