Fermat's last theorem. Basic tools. Translated from the Japanese by Masato Kuwata (Q2842524)

The book at hand is meant as an introduction to the techniques used to proof Fermat's Last Theorem by establishing the modularity of semi-stable rational elliptic curves. It is the first of two books which result from translating one Japanese book and covers roughly one half of the Japanese original.NEWLINENEWLINEFollowing an instructive introduction in which all the main players show up very briefly, the book consists of seven chapters. In each of these the algebraic points of view are stressed more than other possible geometric resp. analytic facts.NEWLINENEWLINENEWLINEThe first chapter introduces the concept of elliptic curves, their behaviour under reduction and their Tate modules.NEWLINENEWLINENEWLINEThe second one explains the main features of modular forms, in particular newforms of level 2 for congruence subgroups of the modular group, and even more specific normalized simultaneous Hecke eigenforms among these (being called primitive forms in the book). The concept of modular curves is predominant, and a good companion for this chapter is the book on modular forms by \textit{F. Diamond} and \textit{J. Shurman} [A first course in modular forms. Berlin: Springer (2005; Zbl 1062.11022)].NEWLINENEWLINENEWLINEThe third chapter is devoted to Galois representations and their properties, and in this context finite flat group schemes are being discussed.NEWLINENEWLINENEWLINEChapter 4 shows that in order to prove the modularity of the semistable rational elliptic curve \(E\) one may replace the mod-3-representation by the mod-5-representation if the first one is not irreducible. Here Mazur's list of possible torsion subgroups of the Mordell-Weil group \(E({\mathbb Q})\) is used.NEWLINENEWLINENEWLINEThe last three chapters deal with deformations of Galois representations and give a criterion under which circumstances the (\(p\)-adic) Hecke algebra will be the universal (\(p\)-adic) deformation ring and provide some background from commutative algebra. This is one of the central points in proving the Shimura-Taniyama conjecture.NEWLINENEWLINESeveral appendices cover topics from arithmetic geometry which are helpful for understanding the text. The literature is structured in a way making explicit which sources are relevant to which topics.NEWLINENEWLINENEWLINEThis book can serve as a introduction to the world of modularity results and will prove valuable for anyone willing to invest some work -- which of course one has to do in order to understand interesting mathematics. In the opinion of the reviewer the author found a good balance between unavoidable omissions and desirable contents of a book like this.