In this article, the author reports on a programme to prove that continuous, odd, irreducible representations \(\rho\) of \(G_{\mathbb{Q}}\), the absolute Galois group of \(\mathbb{Q}\), into \(GL_2({\mathbb{C}})\), arise from holomorphic forms of weight one for congruence subgroups of \(SL_2({\mathbb{Z}})\). This is a case of what is known as the ‘strong Artin conjecture”. This programme has had much success. For the latest the reader can look up the preprints “On icosahedral Artin representations”, by K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor and “On icosahedral Artin representations. II” by R. Taylor both available at the author’s homepage.
For the entire collection see [Zbl 0889.00012].