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E. Artin conjectured in [3]
that the *L*-series associated to any continuous irreducible
representation
,
with *n*>1, is entire.
Recent exciting work of Taylor and others suggests that a complete proof
of Artin's conjecture, in the case when *n*=2 and
is odd,
is on the horizon.
This case of Artin's conjecture is known when the image
of
in
is solvable
(see [27]), and in infinitely
many cases when the image of
is not solvable (see [7]).
In 1998, K. Buzzard suggested a way to combine the
main theorem of [8], along with
a computer computation, to deduce modularity of certain
icosahedral Galois representations. Buzzard and I recently obtained
the following theorem.

**Theorem 6**
The icosahedral Artin representations of conductor

are modular.

We expect our method to yield several more examples.
These ongoing computations are laying a small part of the technical
foundations necessary for a full proof of the Artin conjecture
for odd two dimensional ,
as well as stimulating
the development of new algorithms for computing with modular forms
using modular symbols in characteristic .

*William A. Stein*

*1999-12-01*