Question

What are the crystalline realizations of Artin motives?

In a paper by Kisin and Wortmann, "A note on Artin motives" (google it and you'll find it immediately), they define a suitable category of motives incorporating crystalline realization in analogy with Deligne's absolute Hodge cycles. However, they don't say explicitly what these realizations are (or at least I couldn't find it). Or can I find it somewhere else?

Answer 1

An Artin motive is just the same thing as a continuous representation of the absolute Galois group $G_K$ (of whatever number field $K$ we are thinking about) with finite image.
For conreteness, let's write it as $G_K \to GL(V)$, where $V$ is a finite dimensional vector space over $\mathbb Q$. (We could incorporate coefficients into the picture, by having $V$ be an $E$-vector space over some other number field $E$, but this doesn't really change anything: since the construction of realizations is functorial, the $E$-action will just come along for the ride.)

Our Artin motive will have a crystalline realization at those place $v$ where the representation is unramified. If $v$ is such a place, lying over the prime $p$, then we can consider $\mathbb Q_p \otimes_{\mathbb Q} V$; this is now a $p$-adic representation of $G_K$, unramified at $v$. Restrict it to $G_{K_v}$. Fontaine's theory attaches an $n$-dimensional $K_v$-vector space to $\mathbb Q_p\otimes_{\mathbb Q} V$, its crystalline Dieudonne module, which is equipped with a $\sigma$-linear Frobenius operator; this is the crystalline realization. (One could define it more geometrically, via crystalline cohomology, but would get the same answer.)

In this simple context, one can compute the crystalline Dieudonne module in the following concrete fashion: begin with $\mathbb Q_p\otimes_{\mathbb Q} V$, and give it a "relative Frobenius" operator by looking at the action of the geometric Frobenius at $v$ arising from our given $G_K$-action (restricted to $G_{K_v}$).

This is a linear operator, which will (by construction) turn out to be the $d$th power of the crystalline Frobenius (if $d = [K_v^0:\mathbb Q_p]$, where $K_v^0$ is the maximal subfield of $K_v$ that is unramified over $\mathbb Q_p$).

Now apply a $\sigma$-linear induction (from $\mathbb Q_p$ to $K_v^0$) to $\mathbb Q_p \otimes_{\mathbb Q} V$, to get an $n$-dimensional vector space over $K_v^0$ with a $\sigma$-linear "absolute Frobenius", whose $d$th power is (the extension of scalars from to $K_v^0$ of) the "relative Frobenius" introduced above. This is the crystalline Dieudonne module, or, equivalently, the crystalline realization.