Ramified deformation problems
As is well known, the modularity of any semistable elliptic curve \(E\) over the rationals has been proved by Wiles (and Taylor) in 1994. In the meantime it has been revealed that one can do without assuming semistability [cf. \textit{C. Breuil, B. Conrad, F. Diamond} and \textit{R. Taylor}, J. Am. Math. Soc. 14, 843-939 (2001; Zbl 0982.11033)]. As a vital intermediate step, these authors use results from \textit{B. Conrad, F. Diamond} and \textit{R. Taylor} [J. Am. Math. Soc. 12, 521-567 (1999; Zbl 0923.11085)], where we find in particular that \(E\) is modular if it acquires good reduction over a tame extension of \({\mathbb Q}_3\). In the proof of this assertion a new (in comparison to Taylor/Wiles) deformation problem for Galois representations comes up for which one requires the results of the article under review. To be a bit more precise, let \(\overline\rho: \text{Gal}(\overline{\mathbb Q}_p /{\mathbb Q}_p) \rightarrow \text{GL}_2({\mathbb F}_p)\) be the representation on the \(p\)-torsion of an elliptic curve \(E/{\mathbb Q}_p\) that has supersingular reduction over some finite extension \(K\) of \({\mathbb Q}_p\). To this representation there corresponds a certain deformation problem (depending on \(K\) and a mixed characteristic, complete, discrete valuation ring \({\mathcal O}\) with residue field \({\mathbb F}_p\)), and it is shown that the universal deformation ring of this problem is isomorphic to the ring of power series over \(\mathcal O\) in two variables. This result, in turn, heavily relies on the author's results on Honda systems [Compos. Math. 119, 239-320 (1999; Zbl 0984.14015)]. The author also describes a larger class of representations (containing all the above \(\overline\rho\)) by ``technical conditions'' and classifies its elements. For this larger class, the result on the deformation problem is also valid. Lastly, imposing certain conditions on the determinant of the desired lifts of \(\overline\rho\) leads to a deformation problem which has a universal deformation ring that is isomorphic to the ring of power series over \(\mathcal O\) in one variable.
scientific article; zbMATH DE number 1425208
scientific article