Deformations of Galois representations (Q2768080)

This is a largely extended version of an earlier survey of the author [CMS Conf. Proc. 17, 179-207 (1995); Zbl 0841.11027)]. NEWLINENEWLINENEWLINEThe problem of deforming Galois representations (or looking at ``all'' lifts of a modular representation to reasonable representations in characteristic zero) is of vital importance in modern arithmetic algebraic geometry. The main body of the introduction to this topic, which is under review here, is divided up into the following parts: I. Galois groups and their representations. II. Deformations of representations. III. The universal deformation: existence. IV. The universal deformation: properties. V. Explicit deformations. VI. Deformations with prescribed properties. VII. Modular deformations. VIII. \(p\)-adic families and infinite ferns. NEWLINENEWLINENEWLINEQuite naturally it is impossible to prove every statement in an article like this. The notable (and important) exception is the proof of the representability of the deformation functor, which is proved using Schlessinger's criteria. Apart from references, the author often gives hints at the proofs and otherwise leaves them as exercises. NEWLINENEWLINENEWLINEThe paper is rounded off by three appendices: \textit{Mark Dickinson} gives an alternative proof of representability, \textit{Tom Weston} treats a theorem of Flach describing concrete examples of unobstructed deformation problems, and \textit{Matthew Emerton} introduces the \(p\)-adic geometry of modular curves. We finally find an extensive bibliography (with 161 items). With great precision and care the authors produced a very fine overview over this area, which will prove helpful for novices in deformation theory and can help as a guideline for courses on this topic.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00034].