Fix a prime p≥3 and let E be an algebraic extension of Qp. We construct some Wach modules over OE[π] of rank three and calculate mod-p reductions of the corresponding crystalline representations of Gal(Qp/Qp) over E. The main idea in the construction of some Wach modules is to approximate p-adically and π-adically the conditions of Wach modules by simpler conditions. Yamashita and Yasuda (in preparation) have constructed some Wach modules of rank two by employing some approximation methods. We apply their method to construct some Wach modules of rank three.
Takuya Nishisako (Wed,) studied this question.