Characterizing Abelian Varieties by the Reductions of the Mordell-Weil Group

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Let A be an abelian variety defined over a number field K. If p is a prime of K of good reduction for A, let A(K)ₚ denote the image of the Mordell-Weil group via reduction modulo p. We prove in particular that the size of A(K)ₚ, by varying p, encodes enough information to determine the K-isogeny class of A, provided that the following necessary condition is satisfied: B(K) has positive rank for every non-trivial abelian subvariety B of A. This is the analogue to a result by Faltings of 1983 considering instead the Hasse-Weil zeta function of the special fibers Aₚ.