Galois Representations and a Conjecture of Artin

The Galois group $\Gal (\overline{\mathbf{Q}}/\mathbf{Q})$ is an object of central importance in number theory, and I've often heard that in some sense number theory is the study of this group. A good way to study a group is to study how it acts on various objects, that is, to study its representations.

Endow ${\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$ with the topology which has as a basis of open neighborhoods of the origin the subgroups $\Gal (\overline{\mathbf{Q}}/K)$, where $K$ varies over finite Galois extensions of $\mathbf{Q}$. (Note: This is not the topology got by taking as a basis of open neighborhoods the collection of finite-index normal subgroups of ${\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$.) Fix a positive integer $n$ and let $\GL _n(\mathbf{C})$ be the group of $n\times n$ invertible matrices over $\mathbf{C}$ with the discrete topology.

Definition 14.3.1   A of ${\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$ is a continuous homomorphism

$$\rho:{\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\to \GL _n(\mathbf{C}).$$

For $\rho$ to be continuous means that there is a finite Galois extension $K/\mathbf{Q}$ such that $\rho$ factors through $\Gal (K/\mathbf{Q})$:

$$\xymatrix{ {{\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})}\ar[... ...{\rho}\ar[dr]& &{\GL _n(\mathbf{C})}\\ &{\Gal (K/\mathbf{Q})}\ar[ur]_{\rho'}}$$

For example, one could take $K$ to be the fixed field of $\ker(\rho)$. (Note that continous implies that the image of $\rho$ is finite, but using Zorn's lemma one can show that there are homomorphisms ${\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\to\{\pm 1\}$ with finite image that are not continuous, since they do not factor through the Galois group of any finite Galois extension.)

Fix a Galois representation $\rho$ and a finite Galois extension $K$ such that $\rho$ factors through $\Gal (K/\mathbf{Q})$. For each prime $p\in\mathbf{Z}$ that is not ramified in $K$, there is an element $\Frob _\mathfrak{p}\in\Gal (K/\mathbf{Q})$ that is well-defined up to conjugation by elements of $\Gal (K/\mathbf{Q})$. This means that $\rho'(\Frob _p)\in \GL _n(\mathbf{C})$ is well-defined up to conjugation. Thus the characteristic polynomial $F_p\in\mathbf{C}[x]$ is a well-defined invariant of $p$ and $\rho$. Let

$$R_p(x) = x^{\deg(F_p)}\cdot F_p(1/x) = 1 + \cdots + {\mathrm{Det}}(\Frob _p)\cdot x^{\deg(F_p)}$$

be the polynomial obtain by reversing the order of the coefficients of $F_p$. Following E. Artin, set

$$L(\rho,s) = \prod_{p\text{ unramified}} \frac{1}{R_p(p^{-s})}.$$ (14.2)

We view. $L(\rho,s)$ as a function of a single complex variable $s$. One can prove that $L(\rho,s)$ is holomorphic on some right half plane, and extends to a meromorphic function on all $\mathbf{C}$.

Conjecture 14.3.2 (Artin)   The $L$-series of any continuous representation

$$\Gal (\overline{\mathbf{Q}}/\mathbf{Q})\to\GL _n(\mathbf{C})$$

is an entire function on all $\mathbf{C}$, except possibly at $1$.

This conjecture asserts that there is some way to analytically continue $L(\rho,s)$ to the whole complex plane, except possibly at $1$. (A standard fact from complex analysis is that this analytic continuation must be unique.) The simple pole at $s=1$ corresponds to the trivial representation (the Riemann zeta function), and if $n\geq 2$ and $\rho$ is irreducible, then the conjecture is that $\rho$ extends to a holomorphic function on all $\mathbf{C}$.

The conjecture follows from class field theory for $\mathbf{Q}$ when $n=1$. When $n=2$ and the image of $\rho$ in $\PGL _2(\mathbf{C})$ is a solvable group, the conjecture is known, and is a deep theorem of Langlands and others (see [Lan80]), which played a crucial roll in Wiles's proof of Fermat's Last Theorem. When $n=2$ and the projective image is not solvable, the only possibility is that the projective image is isomorphic to the alternating group $A_5$. Because $A_5$ is the symmetric group of the icosahedron, these representations are called . In this case, Joe Buhler's Harvard Ph.D. thesis gave the first example, there is a whole book [Fre94], which proves Artin's conjecture for 7 icosahedral representation (none of which are twists of each other). Kevin Buzzard and I (Stein) proved the conjecture for 8 more examples. Subsequently, Richard Taylor, Kevin Buzzard, and Mark Dickinson proved the conjecture for an infinite class of icosahedral Galois representations (disjoint from the examples). The general problem for $n=2$ is still open, but perhaps Taylor and others are still making progress toward it.