Abstract Let X be a Calabi–Yau threefold defined over an algebraically closed field k of characteristic p>0. It is natural to expect that X has some properties which cannot be seen in characteristic zero. To observe such phenomena, we study Calabi–Yau threefolds constructed from fiber products of elliptic and quasi-elliptic rational surfaces over P 1 . These Calabi–Yau threefolds turn out to be unirational, have fibrations which are not generically smooth. We shall also observe their Artin–Mazur formal groups and the Hodge duality.