Icosahedral Galois representations

E. Artin conjectured 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. 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.

By combining the main result of a recent paper of K. Buzzard and Taylor with a computer computation, Buzzard and I recently proved that the icosahedral Artin representations of conductor  $1376=2^5\cdot 43$ are modular. If I can extend a congruence result of J. Sturm, then our method will yield several more examples. These ongoing computations are laying a part of the foundation 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 in characteristic $\ell$.