Arithmetic Geometry and Number Theory RTG Seminar: The Langlands-Kottwitz-Scholze method for Shimura varieties of abelian type

Seminar | October 15 | 3:10-5 p.m. | 784 Evans Hall

 Alex Youcis, UC Berkeley

 Department of Mathematics

The local (and global) Langlands conjectures attempt to bridge the major areas of harmonic analysis and number theory by forming a correspondence between representations which naturally appear in both areas. A key insight due to Langlands and Kottwitz is that one could attempt to understand such a conjectural correspondence by comparing the traces of natural operators on both sides of the bridge. Moreover, it was realized that Shimura varieties present a natural means of doing this. For global applications, questions of reduction type (at a particular prime $p$) for these Shimura varieties can often be avoided, and for this reason the methods of Langlands and Kottwitz focused largely on the setting of good reduction. But, for local applications dealing with the case of bad reduction is key. The setting of bad reduction was first dealt with, for some simple Shimura varieties. Harris and Taylor then used this, together with the work of many other mathematicians, to prove the local Langlands conjecture for $GL_n$. A decade later Scholze gave an alternative, more geometric, way to understand the case of bad reduction for certain Shimura varieties and was able to reprove the local Langlands conjecture for $GL_n$ in simpler terms. In this talk we will discuss an extension of the ideas of Scholze to a wider class of Shimura varieties, as well as the intended application of these ideas to the local Langlands conjectures for more general groups.