Shimura varieties of abelian type) which, in characteristic zero, have a description of their point sets similar to (\[eqn:LRconj-ver1\]) with the parameter $\phi$ being an abelian motive (constructed using absolute Hodge cycles). For ${\bar{\mathbb{F}}_{p}}$-points, the parameter $\phi$, called an *admissible morphism*, is to represent “a motive over ${\bar{\mathbb{F}}_{p}}$ with $G$-structure”. A proper definition of such object involves a Tannakian-theoretic description of the category of motives. But, the Tannakian category of Grothendieck motives over ${\bar{\mathbb{F}}_{p}}$, being non-neutral, is identified (after choice of a fiber functor) with the representation category of a certain Galois gerb (“motivic Galois gerb”), not an affine group scheme, and “a motive over ${\bar{\mathbb{F}}_{p}}$ with $G$-structure” should be a morphism from this motivic Galois gerb to the neutral Galois gerb attached to $G$.

### From Langlands-Rapoport conjecture to Kottwitz formula {#subsubsec:fromLRtoKF}

The conjectural description (\[eqn:LRconj-ver1\]) allows one to obtain a manageable description of the fixed point set of any Hecke correspondence twisted by a Frobenius $\Phi^m$, and eventually a purely group-theoretic formula for its cardinality; in particular, it gives a formula for the cardinalities of the finite sets ${\mathscr{S}}_{{\mathbf{K}}}(G,X)({\mathbb F}_{q^m})=[{\mathscr{S}}_{{\mathbf{K}}}(G,X)({\bar{\mathbb{F}}_{p}})]^{\Phi^m=\mathrm{id}}$ for each finite extension ${\mathbb F}_{q^m}$ of the base field ${\mathbb F}_q=\kappa(\wp)$ whose knowledge amounts to that of 