Potential modularity -- a survey
Since the proof of the Taniyama-Shimura conjecture by \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 443--551 (1995; Zbl 0823.11029)] and \textit{R. Taylor} and \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11030)], `modularity' and `modularity lifting' results has become one of the most central objects of interest in mathematics. The author provides a chronological account of the developments in this domain starting off with the work of Wiles and Taylor-Wiles. He primarily aims at the (potential) modularity theorems for elliptic curves but also touches upon \textit{J.-P. Serre}'s modularity conjecture [Duke Math. J. 54, 179--230 (1987; Zbl 0641.10026)] that was proved by \textit{C. Khare} and \textit{J.-P. Wintenberger} [Invent. Math. 178, No. 3, 505--586 (2009; Zbl 1304.11042)] and provides an overview of \textit{M. Kisin}'s revolutionary contributions [Ann. Math. (2) 170, No. 3, 1085--1180 (2009; Zbl 1201.14034); J. Am. Math. Soc. 22, No. 3, 641--690 (2009; Zbl 1251.11045)] to the modularity lifting technology. He finishes the article with the more recent (for the time of the writing of the current article under review) higher dimensional versions (that is, concerning Galois representations of dimension greater than \(2\)) of modularity results, which eventually culminated in the proof of the Sato-Tate conjecture in [\textit{T. Barnet-Lamb} et al., Publ. Res. Inst. Math. Sci. 47, No. 1, 29--98 (2011; Zbl 1264.11044)] for elliptic curves over totally real fields.NEWLINENEWLINEThis article is a beautiful exposition of the themes and ideas that go into the proofs of such statements by one of the most influential figures in the field and as such, the reviewer very highly recommends this article to anyone who wishes to gain a perspective on this very rich and fruitful area of research.NEWLINENEWLINEFor the entire collection see [Zbl 1237.11001].
scientific article; zbMATH DE number 6059398