The Support Problem for Abelian Varieties

Let A be an abelian variety over a number field K. If P and Q are K-rational points of A such that the order of the (mod p) reduction of Q divides the order of the (mod p) reduction of Q for almost all primes (mod p), then there exists a K-endomorphism φ of A and a positive integer k such that kQ = φ(P). In this paper, we solve the support problem for abelian varieties over number fields, thus answering a question of C. Corralez-Rodrigáñez and R. Schoof [3]. Recently, G. Banaszak, W. Gajda, and P. Krasoń [1] and C. Khare and D. Prasad [4] have solved the problem for certain classes of abelian varieties for which the images of the l-adic Galois representations can be particularly well understood. Our result is the following: Theorem 1: Let K be a number field, OK its ring of integers, and O the coordinate ring of an open subscheme of SpecOK . Let A be an abelian scheme over O and P,Q ∈ A(O) arbitrary sections. Suppose that for all n ∈ Z and all prime ideals p of O, we have the implication (1) nP ≡ 0 (mod p) ⇒ nQ ≡ 0 (mod p). Then there exists φ ∈ EndO(A) and k ∈ Z >0 such that