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.

For 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

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 .

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