Arithmetic Moduli at Infinite Level

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The Langlands program endeavors to link Galois representations to automorphic forms. Central to the Langlands program is the study of arithmetic moduli, which is to say parameter spaces for geometric objects defined over a local or global field. Arithmetic moduli include modular curves and Shimura varieties in the global setting, and the Lubin-Tate tower and spaces of Rapoport-Zink in the local setting. Such geometric objects are always structured in towers, such as the tower of modular curves of level a power of p. Taken as a whole, a tower of arithmetic moduli admits an action of a reductive group, and studying this action on the cohomology of the tower is the only way we know how to attach a Galois representation to an automorphic form. This project concerns arithmetic moduli at infinite level, e.g. the inverse limit along a tower of modular curves. A recent discovery of the PI is that, in the case of the Lubin-Tate tower, such limits exist as objects in Peter Scholze's new category of perfectoid spaces. There are many ways in which the inverse limit object is actually simpler than the constituent layers of the tower. The PI intends to generalize these discoveries to general arithmetic moduli. These results will be leveraged into new insights in the Langlands program. In particular this proposal represents the most promising hope yet for a proof of the local Langlands correspondence for GL(n) which is purely local in nature (i.e., which does not involve automorphic representations).This proposal includes a plan for the research-level participation of graduate and undergraduate students at Boston University, with appropriate projects for each. The PI intends to continue his involvement in programs which disseminate mathematics throughout a broad community. These programs include the PROMYS program, a number theory summer program for high school students located in Boston University, and the Arizona Winter School, an intensive mini-course for graduate students which takes place in Tucson. The PI also intends to present research at conferences, including the joint meetings of the AMS-MAA.