Computing the Trace of Frobenius

Abstract

This chapter aims to compute the trace Tr(Frob-1 ¦H* (BunG(X);Zℓ)), where ℓ is a prime number which is invertible in F q. It follows the strategy outlined in Chapter 1. If X is an algebraic curve over the field C of complex numbers and G is a smooth affine group scheme over X whose fibers are semisimple and simply connected, then Theorem 1.5.4.10 (and Example 1.5.4.15) supply a quasi-isomorphism whose right-hand side is the continuous tensor product of Construction 1.5.4.8. The remainder of this chapter is devoted to explaining how Theorem 4.1.2.1 can be used to compute the trace of the arithmetic Frobenius automorphism on the ℓ-adic cohomology of BunG(X).