# Galois Representations and a Conjecture of Artin

The Galois group 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 with the topology which has as a basis of open neighborhoods of the origin the subgroups , where varies over finite Galois extensions of . (Note: This is not the topology got by taking as a basis of open neighborhoods the collection of finite-index normal subgroups of .) Fix a positive integer and let be the group of invertible matrices over with the discrete topology.

Definition 14.3.1   A of is a continuous homomorphism

For to be continuous means that there is a finite Galois extension such that factors through :

For example, one could take to be the fixed field of . (Note that continous implies that the image of is finite, but using Zorn's lemma one can show that there are homomorphisms 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 and a finite Galois extension such that factors through . For each prime that is not ramified in , there is an element that is well-defined up to conjugation by elements of . This means that is well-defined up to conjugation. Thus the characteristic polynomial is a well-defined invariant of and . Let

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

 (14.2)

We view. as a function of a single complex variable . One can prove that is holomorphic on some right half plane, and extends to a meromorphic function on all .

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

is an entire function on all , except possibly at .

This conjecture asserts that there is some way to analytically continue to the whole complex plane, except possibly at . (A standard fact from complex analysis is that this analytic continuation must be unique.) The simple pole at corresponds to the trivial representation (the Riemann zeta function), and if and is irreducible, then the conjecture is that extends to a holomorphic function on all .

The conjecture follows from class field theory for when . When and the image of in 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 and the projective image is not solvable, the only possibility is that the projective image is isomorphic to the alternating group . Because  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 is still open, but perhaps Taylor and others are still making progress toward it.

William Stein 2004-05-06