Galois Representations

The absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is one of the most important objects of study in mathematics. However the direct study of this group is very difficult; for instance it is an infinite group, and we know very little about it. To make it easier for us, we will often instead study representations of this group – i.e. group homomorphisms to the group $\text{GL}(V)$ of linear transformations of some vector space $V$ over some field $F$. When $V$ has finite dimension $n$, $\text{GL}(V)$ is just $\text{GL}_{n}(F)$, the group of $n\times n$ matrices with entries in $F$ and nonzero determinant. Often we will also want the field $F$ to carry a topology – this will also endow $\text{GL}_{n}(F)$ with a topology. For instance, if $F$ is the $p$-adic numbers $\mathbb{Q}_{p}$ it has a $p$-adic topology (see also Valuations and Completions). Since $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ has its own topology, we can then talk about representations which are continuous. In this post we shall consider three examples of these continuous Galois representations.

Our first example of a Galois representation is known as the $p$-adic cyclotomic character. This is a one-dimensional representation over the $p$-adic numbers $\mathbb{Q}_{p}$, i.e. a group homomorphism from $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}$ to $\text{GL}_{1}(\mathbb{Q}_{p})$, which also happens to just be the multiplicative group $\mathbb{Q}_{p}^{\times}$. Let us explain how to obtain this Galois representation.

Consider a primitive $p^{n}$-th root of unity $\zeta_{p^{n}}$. Any element $\sigma$ of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on $\zeta_{p^{n}}$ and sends it to some $p^{n}$-th root of unity, which amounts to raising it to some integer power between $1$ and $p^{n}-1$, i.e. an element of $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$. We now define the $p$-adic cyclotomic character $\chi$ to be the map from $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to $\mathbb{Z}_{p}^{\times}$ which sends the element $\sigma$ to the element of $\mathbb{Z}_{p}^{\times}$ which after modding out by $p^{n}$ is precisely the integer power to which we raised $\zeta_{p^{n}}$.

Our second example of a Galois representation is known as the Tate module of an elliptic curve. We recall that we also discussed an example of a Galois representation coming from the $p$-torsion points of an elliptic curve in Elliptic Curves. The Tate module is a way to package the action of the Galois group not only the $p$-torsion points but also the $p^{n}$-torsion for any $n$, by taking an inverse limit over $n$. Now the $p^{n}$-torsion points are isomorphic to $(\mathbb{Z}/p^{n}\mathbb{Z})^{2}$, so the inverse limit is going to be isomorphic to $\mathbb{Z}_{p}^{2}$. This is not a vector space, since $\mathbb{Z}_{p}$ is not a field, so we take the tensor product with $\mathbb{Q}_{p}$ to get $\mathbb{Q}_{p}^{2}$, which is a vector space. Therefore we get a Galois representation, i.e. a homomorphism from $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to $\text{GL}_{2}(\mathbb{Q}_{p})$. This construction also works for abelian varieties – higher dimensional analogues of elliptic curves – except that the Tate module is now $2g$-dimensional, where $g$ is the dimension of the abelian variety.

Our last example of a Galois representation is given by the $\ell$-adic cohomology (explanation of this terminology to come later) of a smooth proper algebraic variety $X$ over $\mathbb{Q}$. This is the inverse limit over $n$ of the etale cohomology (see also Cohomology in Algebraic Geometry) of $X$ with coefficients in the constant sheaf $\mathbb{Z}/p^{n}\mathbb{Z}$. These etale cohomology groups are somewhat confusingly denoted $H^{i}(X,\mathbb{Z}_{p})$ – note that they are not the etale cohomology of $X$ with $\mathbb{Z}_{p}$ coefficients! Just as in the case of the Tate module, we take the tensor product with $\mathbb{Q}_{p}$ to produce our Galois representation.

These Galois representations coming from the $\ell$-adic cohomology somewhat subsume the Tate modules discussed earlier – that is because, if $X$ is an elliptic curve or more generally an abelian variety, we have that the $\mathbb{Q}_{p}$-linear maps from the Tate module (tensored with $\mathbb{Q}_{p}$) is isomorphic to the first $\ell$-adic cohomology $H_{1}(X,\mathbb{Z}_{p})\otimes\mathbb{Q}_{p}$. We say that the first $\ell$-adic cohomology is the dual of the Tate module.

Although we discussed representations over $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ in this post, it is also often useful to make our study “local” and focus on a single prime $\ell$, and study $\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ instead. In this case we might as well just have replaced $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ with $\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ in the above discussion, and nothing really changes, as long as the primes $\ell$ and $p$ are different primes. In the case that they are the same prime, things become much more complicated (and the theory is far richer)!

Note: Usually, when discussing “local” Galois representations, the notation for the primes $p$ and $\ell$ are switched! In other words, our local Galois representations are group homomorphisms from $\text{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ to $\text{GL}_{n}(\mathbb{Q}_{\ell})$. This is the reason for the terminology “$\ell$-adic cohomology”. Since we started out just discussing “global” Galois representations, I switched the notation to use $p$ instead for the only instances were we needed a prime. Hopefully this is not overly confusing. We can also study Galois representations more generally for number fields (“global”) and finite extensions of $\mathbb{Q}_{p}$ (“local”).

Finally, although we stated above that we will only discuss three examples here, let us mention a fourth example: Galois representations can also come from modular forms (see also Modular Forms). To discuss these Galois representations would require us to develop some more machinery first, so we leave this to future posts for now.

References:

Cyclotomic character on Wikipedia

Tate module on Wikipedia

Etale cohomology on Wikipedia

Reciprocity laws and Galois representations: recent breakthroughs by Jared Weinstein