  In this paper we pose and answer the following question in a few different
contexts: Given a homomorphism f:L_1 --> L_2 of a ``lattices'' that ``reduces
mod p'' for almost all primes p, is f ``algebraic''? For instance the lattices
may be the Mordell-Weil lattices of rational points of abelian varieties over
Q, or arithmetic groups etc. Implicit in an affirmative answer to the question
for Mordell-Weil lattices is a novel criterion for abelian varities to be
isogenous.
