Mathematical Sciences: The Integral Hecke Structure of Shimura Curves

这项研究涉及的志村曲线与一个 不定有理四元数除代数 合理 这些曲线的结构已经被理解了很长时间， 然而，有许多重要的问题需要回答， 关于其整体结构。 德林费尔德给了 描述为课程模块方案，本研究将 研究它们的Hecke结构。 这项研究是在最博学的领域之一， 算术代数几何，融合了 代数和几何（在这种情况下是分析）技术， 回答数论中的问题。 这项研究已经 是解决世界上最大的 今年将继续朝着同一方向发展， 这个奖项的任期。 很多人会很感兴趣 完成了

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.