The constant of the support problem for abelian varieties

Abstract

Let A be an abelian variety defined over a number field K and let P and Q be points in A(K) satisfying the following condition: for all but finitely many primes p of K. the order of (Q mod p) divides the order of (P mod p). Larsen proved that there exists a positive integer c such that cQ is in the End(K)(A)-module generated by P. We study the minimal value of c and construct some refined counterexamples. (c) 2013 Elsevier Inc. All rights reserved.