On Drinfeld's universal special formal module
type: Item
instance of: scholarly article
title: On Drinfeld's universal special formal module
zbMATH Open document ID: 1174.14020
published in: Journal of Number Theory
publication date: 2009-05-19T00:00:00Z
review text: 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 …
reviewed by: Matthew Morrow
Mathematics Subject Classification ID: 14G32
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 5554700
zbMATH Keywords: Drinfeld's moduli problem
zbMATH Keywords: special formal module with given period
zbMATH Keywords: higher dimensional \(p\)-adic upper half plane
zbMATH Keywords: period lattice of an abelian variety
author: Bingyong Xie
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.12.011
OpenAlex ID: W2092798948
cites work: Q4026193
cites work: Coverings of p-adic symmetric regions
cites work: Q4884545
cites work: Q3901604
cites work: On Drinfeld's universal formal group over the \(p\)-adic upper half plane
DOI: 10.1016/J.JNT.2008.12.011
description: scientific article; zbMATH DE number 5554700