  Consider groups such as Mordell-Weil groups of abelian varieties over number
fields, odd algebraic $K$-theory groups of number fields, or finitely generated
subgroups of the multiplicative groups of number fields. They are all equipped
with systems of reduction maps; thus, one can investigate the Hasse-Minkowski
theorem for quadratic forms with coefficients in such groups. In this paper, we
prove that the theorem holds for the forms whose rank equals $2$ or $3$, and we
demonstrate that it does not hold for higher ranks by providing a
counterexample. We also show that our results constitute a generalization of
the classic Hasse-Minkowski theorem for binary and ternary integral forms.
