Serre's modularity conjecture and new proofs of Fermat's last theorem (Q2894603)

The author presents the main ideas which led to the proof of \textit{J.-P. Serre}'s conjecture [Duke Math.J., 54, 179--230 (1987; Zbl 0641.10026)] which asserted that every odd continuous irreducible \(2\)-dimensional representation over the algebraic closure of \(F_p\) of the Galois group \(G_Q\) of the algebraic closure of the rationals is modular. In the first part of the paper he recalls the fundamental notions of the theory of modular forms and Galois representations, states both version (weak and strong) of the conjecture, presents a short history of it, and comments on its consequences: the Taniyama-Shimura conjecture, Fermat's Last Theorem, classification of group schemas of type \((p,p)\) J.-P. Serre [loc. cit.], modularity of certain abelian varieties [\textit{K. Ribet}, Prog. Math., 224, 241--261 (2004; Zbl 1092.11029)] and of \(3\)-dimensional rigid Calabi--Yau varieties [\textit{F. Q.Gouvêa} and \textit{N. Yui}, Expo. Math. 29, No. 1, 142--149 (2011; Zbl 1230.14056)], and the non-vanishing of Ramanujan's function \(\tau(n)\) for \(n\leq 2\cdot10^{19}\) [Bosman, preprint in \url{arXiv:0710.1237v1}].NEWLINENEWLINEThe second part of the paper is devoted to a well readable sketch of the proof, first in a simpler special case of level one, and then in the general case. In the last part two generalizations of Serre's conjecture are mentioned. One relates continuous semisimple \(n\)-dimensional representations of \(G_Q\), satisfying certain parity conditions, to Hecke classes [\textit{A. Ash} and \textit{W. Sinnott}, Duke Math. J. 105, No. 1, 1--24 (2000; Zbl 1015.11018)] and [\textit{A. Ash, D. Doud} and \textit{D. Pollack}, Duke Math. J. 112, No. 3, 521--579 (2001; Zbl 1023.11025)], and the second, due to \textit{K. Buzzard, F. Diamond} and \textit{F. Jarvis} [Duke Math. J. 155, No. 1, 105--161 (2010; Zbl 1227.11070)], generalizes Serre's conjecture to two-dimensional representations of the Galois group \(\text{Gal}(\overline K/K)\) of the algebraic closure of a totally real algebraic number field \(K\).