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.
Fix a Galois representation and let be the fixed field of , so 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 of is a well-defined invariant of and . Let
The conjecture is known when . Assume for the rest of this paragraph that is odd, i.e., if is complex conjugation, then . 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 image of in is not solvable, the only possibility is that the projective image is isomorphic to the alternating group . Because 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 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 is in fact now completely solved, due to recent work of Khare and Wintenberger [KW08] that proves Serre's conjecture.
William Stein 2012-09-24