Gekeler, Ernst-Ulrich On the arithmetic of some division algebras. (English) Zbl 0753.11025 Comment. Math. Helv. 67, No. 2, 316-333 (1992). From the pioneering work of Deuring one knows that there is an intimate relationship between elliptic curves in characteristic \(p\) and the arithmetic of the (definite-quaternion) division algebra \(H(p)\). Deuring’s results were based on the class number formula of Eichler whose original proof was analytic. However, a geometric proof may be given by using the 1-1 correspondence between supersingular elliptic curves and left ideal classes in a maximal order in \(H(p)\). In this paper, the author carries out a similar program for function fields using supersingular Drinfeld modules. One obtains explicit expressions for certain class numbers as well as a “mass” formula. Reviewer: D.Goss (Columbus / Ohio) Cited in 6 Documents MSC: 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11R58 Arithmetic theory of algebraic function fields 11R52 Quaternion and other division algebras: arithmetic, zeta functions Keywords:maximal orders; mass formula; division algebra; function fields; supersingular Drinfeld modules; class numbers PDFBibTeX XMLCite \textit{E.-U. Gekeler}, Comment. Math. Helv. 67, No. 2, 316--333 (1992; Zbl 0753.11025) Full Text: DOI EuDML