On the modularity of certain 2-adic Galois representations
Fix a rational prime \(\ell\), and consider an irreducible, continuous, odd representation \(\rho: G_{\mathbb{Q}}\to \text{GL}_2 (\overline{\mathbb{Q}}_\ell)\) of the absolute Galois group of \(\mathbb{Q}\). If \(\rho\) is unramified almost everywhere and potentially semistable at \(\ell\), one expects (Fontaine and Mazur, Langlands) that \(\rho\) arises from a modular form. On the other hand we have a conjecture of Serre that every irreducible, continuous, odd representation \(\overline{\rho}: G_{\mathbb{Q}}\to \text{GL}_2 (\overline{\mathbb{F}}_\ell)\) also arises from a modular form. If \(\ell\) is odd, then Wiles, Taylor (and others) show that for certain \(\rho\) with irreducible \(\text{mod-}\ell\) reduction \(\overline{\rho}\), the modularity of \(\rho\) can be deduced from that of \(\overline{\rho}\). The author establishes analogues of some of their results in the case \(\ell=2\). Let \((K:\mathbb{Q}_2)< \infty\), with ring of integers \({\mathcal O}\) and residue field \(k\). Let \(\overline{\rho}: G_{\mathbb{Q}}\to \text{GL}_2 (k)\) be an absolutely irreducible continuous representation. Let \(S\) be a finite set of odd primes, containing all primes at which \(\overline{\rho}\) is ramified. There is a universal \(S\)-deformation \((R_S^{\text{univ}}, \rho_S^{\text{univ}})\) of \(\overline{\rho}\) (Prop. 2) and the modular \(S\)-deformation \((R_S^{\text{mod}}, \rho_S^{\text{mod}})\) with the surjective map \(\phi_S: R_S^{\text{univ}}\to R_S^{\text{mod}}\). The main result (Theorem 4) says that \(\phi_S\) is an isomorphism and \(R_S^{\text{univ}}\) is a complete intersection ring. The results of this article are used in the proof of the strong Artin conjecture (the case of icosahedral Artin representations) for an infinite class of representations [\textit{K. Buzzard, M. Dickinson, N. Shepherd-Barron} and \textit{R. Taylor}, Duke Math. J. 109, 283-318 (2001; Zbl 1015.11021), see the following review].
scientific article; zbMATH DE number 1820841