Galois Representations, $L$-series and a Conjecture of Artin

The Galois group $\Gal (\overline{\mathbf{Q}}/\mathbf{Q})$ is an object of central importance in number theory, and much of 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 9.5.1   A complex $n$-dimensional representation 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 if $K$ is the fixed field of $\Ker (\rho)$, then $K/\mathbf{Q}$ is finite Galois. We have a diagram

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

Remark 9.5.2   Continuous implies that the image of $\rho$ is finite, but the converse is not true. Using Zorn's lemma, one can show that there are homomorphisms ${\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\to\{\pm 1\}$ with image of order $2$ that are not continuous, since they do not factor through the Galois group of any finite Galois extension.

Fix a Galois representation $\rho$ and let $K$ be the fixed field of $\ker(\rho)$, so $\rho$ 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 $F_p(x)\in\mathbf{C}[x]$ of $\rho'(\Frob _p)$ is a well-defined invariant of $p$ and $\rho$. Let

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

be the polynomial obtain by reversing the order of the coefficients of $F_p$. Following E. Artin [Art23,Art30], set

$$L(\rho,s) = \prod_{p\text{ unramified}} \frac{1}{R_p(p^{-s})}.$$ (9.3)

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 9.5.3 (Artin)   The $L$-function 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 is known 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 image of $\rho$ in $\PGL _2(\mathbf{C})$ is not solvable, the only possibility is that the projective image is isomorphic to the alternating group $A_5$. Because $A_5$ is the symmetry group of the icosahedron, these representations are called icosahedral. In this case, Joe Buhler's Harvard Ph.D. thesis [Buh78] gave the first example in which $\rho$ was shown to satisfy Conjecture 9.5.3. There is a book [Fre94], which proves Artin's conjecture for 7 icosahedral representation (none of which are twists of each other). Kevin Buzzard and the author proved the conjecture for 8 more examples [BS02]. Subsequently, Richard Taylor, Kevin Buzzard, Nick Shepherd-Barron, and Mark Dickinson proved the conjecture for an infinite class of icosahedral Galois representations (disjoint from the examples) [BDSBT01]. The general problem for $n=2$ is still open, but Taylor and others are making amazing progress toward it.