Deforming Galois representations: Controlling the conductor
The author first discusses some basic results of \textit{B. Mazur} [Galois groups over \({\mathbb{Q}}\), Proc. Workshop, Berkeley 1987, Publ., Math. Sci., Res. Inst. 16, 385-437 (1989)] about deformations of Galois representations \({\bar \rho}\): Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\to GL_ n(k)\) (k a finit field of characteristic p) and briefly recalls some facts from Hida's theory of p-adic modular forms and p-adic Hecke- algebras. If \(n=2\) and \({\bar \rho}\) is attached to a p-ordinary newform, he then gives a new proof (using universal deformations) of the existence of a natural map from the p-ordinary deformation ring \(R^ 0\) of \({\bar \rho}\) to the completion \({\mathcal R}^ 0\) of the p-ordinary Hecke algebra at the maximal ideal determined by \({\bar \rho}\). Under some restrictions, the existence of such a map was first shown by Hida, Mazur- Wiles and Mazur-Tilouine. Finally the author points out that if \(n=2\) and \({\bar \rho}\) is attached to a newform, using a result of \textit{H. Carayol} [Ann. Sci. Éc. Norm. Super., IV. Ser. 19, 409-468 (1986; Zbl 0616.10025)] one can easily obtain estimates for the conductor of a deformation \(\rho\) of \({\bar \rho}\) in terms of the conductor of \({\bar \rho}\). This allows, in particular, to prove that under certain conditions the above map \(R^ 0\to {\mathcal R}^ 0\) is surjective (a fact previously proved by Mazur- Tilouine under more restrictive hypotheses).
scientific article; zbMATH DE number 4156543