CHAPTER 5

Frobenius Conjugacy Classes

Let G/k be a form of G_m, and N in P_arith an object which is ι-pure of weight zero and arithmetically semisimple. If G/k is G_m/k, then we have the fibre functor on <N>_arith given by

M !(N) := H⁰(A¹/k, j_0!M),

on which Frob_k operates. And for each finite extension field E/k and each character χ of G(E), we have the fibre functor !_χ on <N>_arith given by

!_χ(M) := !(M ⊗L_χ),

on which Frob_E operates. This action of Frob_E on !_χ gives us an element in the Tannakian group G_arith,N,!x for <N>_arith, and so a conjugacy class Frob_E,χ in the reference Tannakian group G_arith,N := G_arith,N,!. By the definition of this conjugacy class, we have an identity of characteristic polynomials

det(1 – T Frob_E,χ|!(N)) = det(1 – T Frob_E|!_χ(N)).

When G/k is the nonsplit form, then the definition of ! depends upon a choice, and so in general only Frob_k2 acts on it. For a finite extension field E/k and a character χ of G(E), we have the fibre functor !_χ on <N>_arith given by

!_χ(M) := !(M ⊗L_χ),

but we have Frob_E acting on it only if either deg(E/k) is even (in which case G/E is G_m/E) or if χ is good for N, in which case !_χ is the fibre functor

M H⁰_c(G/k, M ⊗L_χ),

on which Frob_E acts. So we get conjugacy classes Frob_E,χ in the reference Tannakian group G_arith,N when either χ is good for N or when E/k has even degree. And again here we have an identity of characteristic polynomials

det(1 – T Frob_E,χ|!(N)) = det(1 – TFrob_E|!_χ(N)).

In either the split or nonsplit case, when χ is good for N, the conjugacy class Frob_E,χ has unitary eigenvalues in every representation of the reductive group G_arith,N. Now fix a maximal compact subgroup K of the complex reductive group G_arith,N (C). As explained in the introduction, the semisimple part (in the sense of Jordan decomposition) of Frob_E,χ gives rise to a well-defined conjugacy class θ_E,χ in K.

As we will see later when we try to compute examples, the Frobenius conjugacy classes Frob_E,χ in G_arith,N attached to χ’s which are not good for N will also play a key role, providing a substitute for local monodromy. But for the time being we focus on the classes θ_E,χ in K.