In practice, how explicitly can we describe a Galois representation?

Question

In most parts of representation theory, we naturally want to describe a given representation as explicitly as possible. This seems to me a feasible project only if, as a first step, the group itself can be explicitly described. This is the case with finite groups, Lie groups, algebraic groups etc. But a group such as the absolute Galois group $G_\mathbf{Q}$ has no explicit description as far as I know.

So how does one in practice go about trying to study representations of $G_\mathbf{Q}$? If we do not explicitly understand the group itself, how can we hope to describe its representations, and in terms of what?

Answer 1

(Too long for a comment, but maybe not what you are looking for.) I think the Galois representations that you are interested in are those coming from geometry with coefficients in a $p$-adic field.

Firstly, they usually come as a sequence of Galois representations that factor through a finite extension. For instance the representation for an abelian variety $A$ is built from the action of the Galois group on the finite torsion subgroups $A[p^k]$. You could now study the tower of fields obtained from adjoining these torsion points.

Though, in my view the best information that we can extract from a Galois representation is local. For each place of the number field, we get a representation of the local Galois group of the completion at this place. The local Galois group has an explicit description. In the geometric setting, for almost all places, this is an unramified representation and so the only information there is the image of the Frobenius. This gives you one matrix for each of the unramified places and you can study the trace of the matrix etc. All these together already give you the $L$-function, which contains a lot of information about your original geometric object. The ramified place, and in particular those above $p$, will contain more information and that is harder to extract but very well studied.

Finally, the Galois representation contains a lot of information, but the associated motive even a little more, like periods.

I would say that for one fixed representation, one usually thinks of a tool that tries to describe the geometric object you started from. If you want to use representations to learn something about the group $G_{\mathbb{Q}}$ you would need to take a whole class of representations at once.

Answer 2

Let $K$ be a perfect field, let $\overline{K}$ be an algebraic closure, and let $G$ be the Galois group of $\overline{K}$ over $K$. Let $\mathcal{E}_K$ be the category of finite étale $K$-algebras, i.e. finite products of finite extension fields of $K$. If $X$ is a continuous $G$-set and $A\in\mathcal{E}_K$ then we can define $$ (FX)(A) = \text{Map}_G(\text{Alg}_K(A,\overline{K}),X) $$ This construction gives an equivalence from continuous $G$-sets to functors $Y\colon\mathcal{E}\to\text{Set}$ with the property that all diagrams of the form $$ Y(A) \to Y(B) \rightrightarrows Y(B\otimes_AB) $$ are equalisers. If $X$ is a vector space with $G$ acting linearly, then $FX$ will be a functor taking values in vector spaces.

Typically we cannot describe $\overline{K}$ or $G$ or $X$ explicitly, but we can describe $FX$. Indeed, in typical examples, if you look closely at the definition of $X$ you will find that it is effectively defined as $F^{-1}Y$ for some fairly explicit $Y$. Thus, one answer to your question is that you should ignore Galois representations and formulate everything in terms of functors of étale algebras instead.