References for Artin motives

Question

I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in MO to provide me a reference which can be more useful to me than this terse description. Any comments and explanations also will be helpful. I apologize for asking a seemingly basic question; but I find it impossible to wade through the numerous references available on motives.

Fix a field $k$ and consider the functor finite separable extensions $K$ of $k$ → finite sets with a (continuous) action of the absolute Galois group of $k$ which maps $K$ to the (finite) set of embeddings of $K$ into an algebraic closure of $k$. In Galois theory this functor is shown to be an equivalence of categories. Notice that fields are $0$-dimensional. Motives of this kind are called Artin motives. By $\mathbb Q$-linearizing the above objects, another way of expressing the above is to say that Artin motives are equivalent to finite $\mathbb Q$-vector spaces together with an action of the Galois group.

I have some idea of what are pure motives and mixed motives, in the context of algebraic varieties. What I exactly have in mind is to understand the modern statement of equivariant Tamagawa number conjecture. This would appear to be the simplest instance to keep in mind, if I go ahead.

Answer 1

André's book is the main reference for the "yoga" of motives. You'll find a description of Artin motives in the Voevodsky formalism in

Beilinson and Vologodsky - http://www.math.uiuc.edu/K-theory/0832/

Wildehaus - http://www.math.uiuc.edu/K-theory/0918/

From the tannakian view point, Artin motives are just representation of the usual Galois group. So, as motives, Artin motives are not that interesting. It's just the usual Galois theory of fields.

Answer 2

A motive is a chunk of a variety cut out by correspondences. (If you like, it is something of which we can take cohomology.)

Artin motives are what one gets by restricting to zero-dimensional varieties. If the ground field is algebraically closed then zero-dimensional varieties are simply finite unions of points, so there is not much to say; the only invariant is the number of points.

But if the ground field $K$ is not algebraically closed (but is perfect, e.g. char $0$, so that we can describe all finite extensions by Galois theory), then there are many interesting $0$-dimensional motives, and in fact the category of Artin motives (with coefficients in a field $F$ of characteristic $0$, say) is equal to the category of continuous representations of $Gal(\overline{K}/K)$ on $F$-vector spaces (where the $F$-vector spaces are given their discrete topoogy; in other words, the representation must factor through $Gal(E/K)$ for some finite extension $E$ of $K$).

Perhaps from a geometric perspective, these motives seem less interesting than others. On the other hand, number theoretically, they are very challenging to understand. The Artin conjecture about the holomorphicity of $L$-functions of Artin motives, which is the basic reciprocity conjecture regarding such motives, remains very wide open, with very few non-abelian cases known. (Of course, for representations with abelian image, these conjectures amount to class field theory, which is already quite non-trivial.)