Reduction of homomorphisms mod p and algebraicity

Let A be an abelian variety over a number field K. Let φ be an endomorphism of A(K) into itself which reduces modulo v for almost all finite places v of K. The question we discuss in this paper is whether φ arises from an endomorphism of the abelian variety A. We answer this question in the affirmative for many cases. The question is inspired by a work of C. Corrales and R. Schoof, and uses a recent work of Larsen. We also look at the analogue of this question for linear algebraic groups.