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