Ramified deformation problems
type: Item
instance of: scholarly article
title: Ramified deformation problems
zbMATH Open document ID: 0997.11042
DOI: 10.1215/S0012-7094-99-09718-1
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: 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 whic…
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 1425208
zbMATH Keywords: deformation problems
zbMATH Keywords: Honda systems
MaRDI profile type: Publication
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full work available at URL: s0012-7094-99-09718-1
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description: scientific article; zbMATH DE number 1425208