Langlands-Kottwitz Method on Moduli Spaces of Global Shtukas

Abstract

We apply the approach of Scholze to compute the trace of Hecke operator twisted by some power of Frobenius on the cohomology of the moduli spaces of global shtukas in the case of bad reduction. We find a formula that involves orbital integrals and twisted orbital integrals which can be compared with the Arthur-Selberg trace formula. This extends the results of Ngo and Ngo Dac on counting points of moduli spaces of global shtukas over finite fields. The main problem lies in finding a suitable compactly supported locally constant function that will be plugged into the twisted orbital integrals. Following Scholze, we construct locally constant functions called the test functions by using deformation spaces of bounded local shtukas. Then we establish certain local-global compatibility to express the trace on the nearby cycle sheaves on the moduli space of global shtukas to the trace on the deformation spaces.