On Drinfeld's universal special formal module
The paper under review studies the higher dimensional \(p\)-adic upper half plane as the moduli space for certain rigidified formal groups, following work by \textit{V.~Drinfel'd} [Funct. Anal. Appl. 10, 107--115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29--40 (1976; Zbl 0346.14010)] and \textit{J.~Teitelbaum} [Math. Ann. 284, No. 4, 647--674 (1989; Zbl 0682.14032)]. Let \(K\) be a non-archimedean local field, with prime \(\pi_K\), and \(d\geq 2\) an integer. The rigid, symmetric space \(\Omega^d\) is defined to be the set of points of \(\mathbb{P}_K^{d-1}\) which do not lie on any hyperplane defined over \(K\). Let \(D\) be the central division algebra over \(K\) with invariant \(1/d\); let \(\mathcal{O}_D\) be the maximal order of \(D\). Let \(\widehat{K^{\text{ur}}}\) be the completion of the maximial unramified extension of \(K\), and let \(B\) be an \(\mathcal{O}_K\)-algebra. A \textit{formal \(\mathcal{O}_D\)-module over \(B\)} is a \(d\)-dimensional, formal group \(G\) over \(B\), together with an embedding \(\mathcal{O}_D\hookrightarrow\text{End}(G)\). If an additional representation theoretic condition is satisfied then \(G\) is said to be \textit{special}. A central result of Drinfel'd's [loc. cit.] is that \(\Omega^d\) is the generic fibre of a formal scheme \(\widehat{\Omega}^d\) over \(\mathcal{O}_K\) which solves the following moduli problem: for any \(\mathcal{O}_{\widehat{K^{\text{ur}}}}\)-algebra \(B\) in which \(\pi_K\) is nilpotent, the \(B\)-valued points of \(\widehat{\Omega}^d\widehat{\otimes}\mathcal{O}_{\widehat{K^{\text{ur}}}}\) naturally correspond to (isomorphism classes of) pairs \((G,\rho)\), where \(G\) is a special, formal \(\mathcal{O}_D\)-module over \(B\) and \(\rho\) is so-called rigidifying data. Moreover, Drinfel'd shows how to explicitly construct the period, i.e. the \(B\)-valued point, associated to the rigidified module. The subject of the paper under review is the inverse procedure, namely to construct the formal module \(G\) and rigidifying data \(\rho\) from the period. The paper begins with a reminder of Drinfel'd's work. Then, given a \(\widehat{K^{\text{ur}}}\)-valued point of \(\Omega^d\), the reduction of the corresponding rigidified, special, formal \(\mathcal{O}_D\)-module is constructed. Using the technique of the universal extension due to \textit{M.~Hazewinkel} [Formal groups and applications. Pure and Applied Mathematics, 78. New York-San Francisco-London: Academic Press (1978; Zbl 0454.14020)], the module itself is then obtained. In the case \(d=2\), these results are originally due to Teitelbaum [loc. cit], who worked with \(K=\mathbb{Q}_p\), though it is remarked in the paper under review that his methods are valid for arbitrary \(K\).
higher dimensional \(p\)-adic upper half plane
scientific article; zbMATH DE number 5554700