Taniyama-Shimura 2: Galois Representations

Fix some proper variety {X/\mathbb{Q}}. Our goal today will seem very strange, but it is to explain how to get a continuous representation of the absolute Galois group of {\mathbb{Q}} from this data. I’m going to assume familiarity with etale cohomology, since describing Taniyama-Shimura is already going to take a bit of work. To avoid excessive notation, all cohomology in this post (including the higher direct image functors) are done on the etale site.

For those that are intimately familiar with etale cohomology, we’ll do the quick way first. I’ll describe a more hands on approach afterwards. Let {\pi: X\rightarrow \mathrm{Spec} \mathbb{Q}} be the structure morphism. Fix an algebraic closure {v: \mathrm{Spec} \overline{\mathbb{Q}}\rightarrow \mathrm{Spec}\mathbb{Q}} (i.e. a geometric point of the base). We’ll denote the base change of {X} with respect to this morphism {\overline{X}}. Suppose the dimension of {X} is {n}.

Let {\ell} be a prime. We consider the constructible sheaf {R^n\pi_*(\mathbb{Z}/\ell^m)}. Now we have an equivalence of categories between these sheaves and continuous {G=Gal(\overline{\mathbb{Q}}/\mathbb{Q})}-modules by taking the stalk at our geometric point. Thus {R^n\pi_*(\mathbb{Z}/\ell^m)_v\simeq H^n(\overline{X}, \mathbb{Z}/\ell^m)} has a continuous action of {G} on it, and hence we get a continuous representation {\rho_{X,m}: G\rightarrow Aut(H^n(\overline{X}, \mathbb{Z}/\ell^m)\simeq GL_d(\mathbb{Z}/\ell^m)}. These all form a compatible family and hence we can take the inverse limit and tensor with {\mathbb{Q}_\ell} to get what is known as an {\ell}-adic Galois representation {\rho_X: G\rightarrow GL_d(\mathbb{Q}_\ell)}. For a technicality that will come up later, we will abuse notation and now relabel {\rho_X} to be the dual (or contragredient) representation.

If you aren’t comfortable with etale cohomology, then you can just use it as a black box cohomology theory to get the same thing as follows. First take the base change {\overline{X}\rightarrow \mathrm{Spec} \overline{\mathbb{Q}}}. Given any element of the Galois group {\sigma \in G} we get an automorphism of {\overline{\mathbb{Q}}}. Thus we can fill in the diagram:

{\begin{matrix} \overline{X} & \stackrel{\sigma}{\rightarrow} & \overline{X} \\ \downarrow & & \downarrow \\ \mathrm{Spec} \overline{\mathbb{Q}} & \stackrel{\sigma}{\rightarrow} & \mathrm{Spec} \overline{\mathbb{Q}} \end{matrix}}

Since {\sigma} was an automorphism, then only thing you have to believe about cohomology is that you then get an isomorphism via pullback {H^n(\overline{X}, \mathbb{Q}_\ell)\stackrel{\sigma^*}{\rightarrow} H^n(\overline{X}, \mathbb{Q}_\ell)}. Thus we get a continuous group homomorphism {G\rightarrow Aut(H^n(\overline{X}, \mathbb{Q}_\ell))} as before. Again, we’ll actually use the dual of this in the future.

To return to an elliptic curve {E} over {\mathbb{Q}}, we know that these are just tori, and hence the first Betti number is {2}. In this case we get that our Galois representation {\rho_E: G\rightarrow GL_2(\mathbb{Q}_\ell)}. If you’ve seen Taniyama-Shimura explained before, this should look familiar. This turns out to be exactly the same representation as the one you get from the Galois action on the Tate module. But the definition of the Tate module requires a group law, and hence the ability to get such a representation doesn’t generalize to all varieties in the way that using middle \ell-adic cohomology does. This is the standard modern approach to defining modularity for other types of varieties.


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