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

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All of our results are stated for 2-dimensional modules with action by the quaternion division algebra overQ p . Drinfeld's results are true in much greater generality. We remark that our results generalize easily to the case of 2-dimensional modules with action by quaternion algebras over extensions ofQ p by applying the theory of formal $$\mathcal{O}$$ -modules. We suspect that Drinfeld's higher dimensional modules over $$W(\bar F_p )$$ are determined by formulae similar to that in Theorem 46, but with α and β generalized to moduli for higher dimensionalQ p -subspaces of $$\hat Q_{p}^{ur} $$ ; however, we have not investigated this in any detail. Although this work amplifies Drinfeld's original paper by supplying many details in certain cases, it is seriously limited in that it considers lifts of SFD modules to unramified rings only. The most interesting points in thep-adic upper half plane are the points defined over ramified rings, which reduce modp to the singular points on the special fiber. What happens there? We do not have a simple answer. Drinfeld's moduli for formal groups on thep-adic upper half plane is the basis for his proof that Shimura curves havep-adic uniformizations. In a later work, we hope to exploit improved versions of the techniques in this work to better understand the arithmetic of Shimura curves. In particular, in the course of work onp-adicL-functions, we have been led to construct certain “p-adic periods” associated to the cohomology of sheaves on Shimura curves which depend essentially on the existence of ap-adic uniformization. We hope to use Drinfeld's moduli to obtain a more natural construction of these periods in terms of the Gauss-Manin connection, and thereby to gain a better understanding of how they might come to appear in special values ofp-adicL-functions.