On the arithmetic of some division algebras (Q1199248)

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.