William Stein's homepage

Icosahedral Galois representations

E. Artin conjectured in [3] that the L-series associated to any continuous irreducible representation $\rho:G_\mathbf{Q}\rightarrow\mbox{\rm GL}_n(\mathbf{C})$, 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 $\rho$ is odd, is on the horizon. This case of Artin's conjecture is known when the image of $\rho$ in $\mbox{\rm PGL}_2(\mathbf{C})$ is solvable (see [27]), and in infinitely many cases when the image of $\rho$ 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  $1376=2^5\cdot 43$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 $\rho$, as well as stimulating the development of new algorithms for computing with modular forms using modular symbols in characteristic $\ell$.