On Drinfeld's universal formal group over the p-adic upper half plane

In his important paper "Coverings of p-adic symmetric regions" [Dr], Drinfeld showed that the p-adic upper half plane and its higher dimensional analogues serve as moduli spaces for certain rigidified formal groups with quaternionic multiplications. Given a formal group of the proper type, together with rigidifying data, over, say, a ring R on which p is nilpotent, Drinfeld constructs an R-valued point of the appropriate p-adic half space. If we draw an analogy between formal groups and abelian varieties, then Drinfeld's procedure is analogous to computing the period lattice of an abelian variety and obtaining thereby a point in a Siegel upper half space. In the case of abelian varieties, the inverse procedure ts well known--Dven a period lattice, the abelian variety can be constructed immediately. The purpose of this paper is to supply the corresponding inverse procedure in the case of Drinfeld's formal groups, at least in a special case. More precisely, given a W(F'p)-valued point of the p-adic upper half plane over Zp, we construct the corresponding 2dimensional, height 4 quaternionic module over W(F,) with its rigidifying data. We proceed in three stages. First, we discuss in detail Drinfeld's functorial description of points on the p-adic upper half plane. We supply a proof of his description, which he states without proof in [Dr]. We also obtain a rigid analytic interpretation of his description. Next, we construct the special fiber of the formal quaternionic module which corresponds to a point on the reduction mod p of the padic upper half plane. Finally, we apply the technique of the universal extension, as described in [Haz], to obtain our desired module over W(F'p). We supply the Dieudonne module and the logarithm for this module in Theorems 45 and 46.