This research concerns the Shimura curve associated to an indefinite rational quaternion division algebra. The rational structure of these curves have been understood for a long time, however there are many important questions to be answered concerning their integral structure. Drinfeld gave the description as a course moduli scheme and this research will investigate their Hecke structure. This research is in one of the most erudite areas of arithmetic algebraic geometry, that subject which blends algebraic and geometric (and in this case analytic) techniques to answer questions in number theory. This research has already been a crucial ingredient in solving one of the big problems settled this year and will continue in the same direction during the tenure of this award. Much of great interest will be accomplished.
