Invitation to the mathematics of Fermat-Wiles.
(Invitation aux mathématiques de Fermat-Wiles.)

*(French)*Zbl 0887.11003
Enseignement des Mathématiques. Paris: Masson. vii, 397 p. (1997).

For the last 350 years, mathematicians have been intrigued by the problem of finding a proof of Fermat’s Last Theorem (FLT), which asserts that the equation \(x^n+ y^n= z^n\), where \(n\geq 3\), has no trivial integer solution. During the period 1982–86, G. Frey, J.-P. Serre and K. Ribet made significant progress in proving that FLT is a consequence of the Shimura-Taniyama conjecture for semistable elliptic curves over \(\mathbb Q\). The proof of this result was given only recently by A. Wiles, in 1994 [Ann. Math. (2) 141, No. 3, 443–551 (1995; Zbl 0823.11029) and A. Wiles and R. Taylor [ibid., 553–572 (1995; Zbl 0823.11030)]. The book under review is an excellent introduction to Wiles’ theory. It contains all the topics which are necessary for the understanding of Wiles’ result: elliptic functions, field theory, elliptic curves, modular functions. Furthermore, it describes the way that Wiles’ theorem implies FLT and shows that the study of FLT is equivalent to the study of the torsion points of ABC-elliptic curves.

The text is well established and accessible for beginners. The treatment is rigorous but is motivated by discussion and examples, and lightened by the inclusion of historical material. Every chapter contains many guides to further study, which will stimulate the reader to delve into the literature devoted to the subject. Moreover, the book contains many carefully selected exercises and interesting problems.

Chapter 1 is of historical nature. The first sections describe the infinite descent method of Fermat, and the proof of FLT for \(n=4\) is given by means of this technique. Section 6 contains a proof of Fermat’s two squares theorem using the arithmetic of the ring \(\mathbb Z[i]\). Section 7 is devoted to Euler’s proof of FLT for \(n=3\). The last sections of chapter 1 contain the proof of the classical result of Kummer which asserts that FLT is valid in the case where \(n\) is a regular prime.

The theory of elliptic functions is described in chapter 2 assuming a familiarity with the basic principles of complex analysis. The first three sections give some ideas on the circumstances of the birth of elliptic functions. Sections 4 and 5 are devoted to the construction of the field of elliptic functions given by Weierstraß and sections 6, 7, and 8 study the Weierstraß cubic. The remaining sections contain another construction of elliptic functions using loxodromic functions, which is the equivalent of Weierstraß’ construction.

Chapter 3 deals with several subjects. The first three sections are devoted to absolute values, completions of valued fields and \(p\)-adic numbers. Sections 4 and 5 contain some generalities about the algebraic closure of a field and the linear representations of groups. The last sections describe the classical Galois theory.

The basic theory of elliptic curves is presented in chapter 4. The first two sections contain some generalities about projective plane curves and the intersection number of two plane curves at a point. Bézout’s theorem is also stated. In section 3 the theorem of nine points is given, which is used in the next section for the description of the group law on an elliptic curve. Section 5 studies the reduction of elliptic curves defined over \(\mathbb Q\) modulo a prime number. The next two sections study the group \(T_N(E)\) of the \(N\)-division points of an elliptic curve \(E\) and, in the case where \(E\) is defined over \(\mathbb Q\), the action of the Galois group \(\text{Gal}(\overline {\mathbb Q}/\mathbb Q)\) on \(T_N(E)\). Section 8 deals with the isogenies and endomorphisms of elliptic curves. In section 9, the proof of Hasse’s theorem about the number of points of an elliptic curve over a finite field is given. Section 10 is devoted to the proof of the Nagell-Lutz theorem, which asserts that the torsion points \((x,y)\) of an elliptic curve \(y^2= f(x)\), where \(f(x)\) is a monic polynomial with integer coefficients, satisfy \(x,y\in\mathbb Z\) with \(y=0\) or \(y\) divides the discriminant of \(f(x)\). The famous Mordell-Weil theorem, which asserts that the group of points of an elliptic curve defined over a number field is finitely generated, is stated without proof in section 11. In section 12 a characterization of elliptic curves is given as curves \(C\) having a simple point \(O\) such that for every positive integer \(n\) the dimension of the space of functions on \(C\) with a pole at \(O\) of order \(\leq n\) is \(n\). The modular invariant of an elliptic curve is introduced in section 13 and it is proved that two elliptic curves \(E\) and \(E'\) over a field \(k\) have the same modular invariant if and only if they are isomorphic over the algebraic closure of \(k\). Section 14 is devoted to the minimal Weierstraß equations of elliptic curves and Neron’s theorem is proved, which asserts that for every elliptic curve over \(\mathbb Q\) there exists a global minimal Weierstraß model. The last section deals with the Riemann zeta function, the zeta function of Artin and the \(L\) function of Hasse-Weil.

Chapter 5 is devoted to the theory of modular forms. In section 1 a historical introduction is given describing some results of Euler and Jacobi on partitions of integers. The next section introduces theta functions and studies some relations between them, their expansion to infinite products, the heat equation and the Jacobi transformation. Section 3 develops the modular properties of Eisenstein series and of the group \(G=\mathrm{SL}_2 (\mathbb Z)/ \{\pm 1\}\), and introduces the modular forms for the congruence groups of level \(N\). In section 4 it is proved that the modular invariant \(j\) is a modular function and defines a bijection from the space of orbits \(H/G\) (of the action of \(G\) on \(H= \{z\in\mathbb C \mid \text{Im}\,z>0\})\) onto \(\mathbb C \cup \{\infty\}\). It follows that an arbitrary nonsingular Weierstraß cubic over \(\mathbb C\) can be parametrized by elliptic functions. The next section studies the vector spaces \(M_k\) and \(S_k\) of modular and cusp forms respectively of weight \(k\) for the group \(G\) and gives some estimations for the coefficients of the expansions of modular and cusp forms. In section 6 it is proved that the space \(S_{2k} (G')\) of modular forms of weight \(2k>0\) for a congruence subgroup \(G'\) of \(G\) is a Hilbert space of finite dimension for the Petersson inner product. Section 7 deals with the Hecke operators for \(\mathrm{SL}_2 (\mathbb Z)\) and the properties of Hecke forms of \(M_k\). The \(L\)-function \(L(f,s)\) of a modular form \(f\) for \(\mathrm{SL}_2(\mathbb Z)\) is studied in the next section and a proof for the functional equation of \(L(f,s)\) is given. The last section contains the conjectures of Hasse and Taniyama-Weil and the statement of Wiles’ theorem.

The principal aim of chapter 6 is to describe that FLT is a consequence of Wiles’ theorem. In the first section, the ring of \(p\)-adic integers is defined as the projective limit of the system of maps \(\mathbb Z/p^n\mathbb Z \to\mathbb Z/p^{n-1}\mathbb Z\). Sections 2 and 3 deal with the Tate module, and the Tate cubics are introduced in section 4. In section 5 it is shown that every nontrivial point \((a,b,c)\) of the Fermat curve \(x^p+y^p +z^p=0\) \((p\) prime \(\geq 5)\), gives rise to the cubic \(E_{a^p, b^p, c^p}: y^2= x(x-a^p) (x-b^p)\), which is a semistable elliptic curve such that the Galois representation of \(\operatorname{Aut}(\overline{\mathbb Q})\) into the Tate module of \(E_{a^p,b^p,c^p}\) at \(p\) is unramified outside of \(2p\). In section 6 Serre’s conjecture on level reduction for modular Galois representations is stated. The next section describes the Mazur-Ribet theorem, which proves a special version of this conjecture and shows that FLT is a consequence of Wiles’ theorem. Furthermore, some other applications are discussed concerning Dénes’ conjecture and the equation \(x^p+y^p +2z^p=0\). The last section presents the famous conjectures of Szpiro and \(abc\), and describes some of their consequences.

The last part of the book is an appendix which gives an historical description of the results, showing that the study of the nontrivial solutions of Fermat’s equation lead to the study of the torsion of the elliptic curve \(E_{A,B,C}\) and conversely.

The text is well established and accessible for beginners. The treatment is rigorous but is motivated by discussion and examples, and lightened by the inclusion of historical material. Every chapter contains many guides to further study, which will stimulate the reader to delve into the literature devoted to the subject. Moreover, the book contains many carefully selected exercises and interesting problems.

Chapter 1 is of historical nature. The first sections describe the infinite descent method of Fermat, and the proof of FLT for \(n=4\) is given by means of this technique. Section 6 contains a proof of Fermat’s two squares theorem using the arithmetic of the ring \(\mathbb Z[i]\). Section 7 is devoted to Euler’s proof of FLT for \(n=3\). The last sections of chapter 1 contain the proof of the classical result of Kummer which asserts that FLT is valid in the case where \(n\) is a regular prime.

The theory of elliptic functions is described in chapter 2 assuming a familiarity with the basic principles of complex analysis. The first three sections give some ideas on the circumstances of the birth of elliptic functions. Sections 4 and 5 are devoted to the construction of the field of elliptic functions given by Weierstraß and sections 6, 7, and 8 study the Weierstraß cubic. The remaining sections contain another construction of elliptic functions using loxodromic functions, which is the equivalent of Weierstraß’ construction.

Chapter 3 deals with several subjects. The first three sections are devoted to absolute values, completions of valued fields and \(p\)-adic numbers. Sections 4 and 5 contain some generalities about the algebraic closure of a field and the linear representations of groups. The last sections describe the classical Galois theory.

The basic theory of elliptic curves is presented in chapter 4. The first two sections contain some generalities about projective plane curves and the intersection number of two plane curves at a point. Bézout’s theorem is also stated. In section 3 the theorem of nine points is given, which is used in the next section for the description of the group law on an elliptic curve. Section 5 studies the reduction of elliptic curves defined over \(\mathbb Q\) modulo a prime number. The next two sections study the group \(T_N(E)\) of the \(N\)-division points of an elliptic curve \(E\) and, in the case where \(E\) is defined over \(\mathbb Q\), the action of the Galois group \(\text{Gal}(\overline {\mathbb Q}/\mathbb Q)\) on \(T_N(E)\). Section 8 deals with the isogenies and endomorphisms of elliptic curves. In section 9, the proof of Hasse’s theorem about the number of points of an elliptic curve over a finite field is given. Section 10 is devoted to the proof of the Nagell-Lutz theorem, which asserts that the torsion points \((x,y)\) of an elliptic curve \(y^2= f(x)\), where \(f(x)\) is a monic polynomial with integer coefficients, satisfy \(x,y\in\mathbb Z\) with \(y=0\) or \(y\) divides the discriminant of \(f(x)\). The famous Mordell-Weil theorem, which asserts that the group of points of an elliptic curve defined over a number field is finitely generated, is stated without proof in section 11. In section 12 a characterization of elliptic curves is given as curves \(C\) having a simple point \(O\) such that for every positive integer \(n\) the dimension of the space of functions on \(C\) with a pole at \(O\) of order \(\leq n\) is \(n\). The modular invariant of an elliptic curve is introduced in section 13 and it is proved that two elliptic curves \(E\) and \(E'\) over a field \(k\) have the same modular invariant if and only if they are isomorphic over the algebraic closure of \(k\). Section 14 is devoted to the minimal Weierstraß equations of elliptic curves and Neron’s theorem is proved, which asserts that for every elliptic curve over \(\mathbb Q\) there exists a global minimal Weierstraß model. The last section deals with the Riemann zeta function, the zeta function of Artin and the \(L\) function of Hasse-Weil.

Chapter 5 is devoted to the theory of modular forms. In section 1 a historical introduction is given describing some results of Euler and Jacobi on partitions of integers. The next section introduces theta functions and studies some relations between them, their expansion to infinite products, the heat equation and the Jacobi transformation. Section 3 develops the modular properties of Eisenstein series and of the group \(G=\mathrm{SL}_2 (\mathbb Z)/ \{\pm 1\}\), and introduces the modular forms for the congruence groups of level \(N\). In section 4 it is proved that the modular invariant \(j\) is a modular function and defines a bijection from the space of orbits \(H/G\) (of the action of \(G\) on \(H= \{z\in\mathbb C \mid \text{Im}\,z>0\})\) onto \(\mathbb C \cup \{\infty\}\). It follows that an arbitrary nonsingular Weierstraß cubic over \(\mathbb C\) can be parametrized by elliptic functions. The next section studies the vector spaces \(M_k\) and \(S_k\) of modular and cusp forms respectively of weight \(k\) for the group \(G\) and gives some estimations for the coefficients of the expansions of modular and cusp forms. In section 6 it is proved that the space \(S_{2k} (G')\) of modular forms of weight \(2k>0\) for a congruence subgroup \(G'\) of \(G\) is a Hilbert space of finite dimension for the Petersson inner product. Section 7 deals with the Hecke operators for \(\mathrm{SL}_2 (\mathbb Z)\) and the properties of Hecke forms of \(M_k\). The \(L\)-function \(L(f,s)\) of a modular form \(f\) for \(\mathrm{SL}_2(\mathbb Z)\) is studied in the next section and a proof for the functional equation of \(L(f,s)\) is given. The last section contains the conjectures of Hasse and Taniyama-Weil and the statement of Wiles’ theorem.

The principal aim of chapter 6 is to describe that FLT is a consequence of Wiles’ theorem. In the first section, the ring of \(p\)-adic integers is defined as the projective limit of the system of maps \(\mathbb Z/p^n\mathbb Z \to\mathbb Z/p^{n-1}\mathbb Z\). Sections 2 and 3 deal with the Tate module, and the Tate cubics are introduced in section 4. In section 5 it is shown that every nontrivial point \((a,b,c)\) of the Fermat curve \(x^p+y^p +z^p=0\) \((p\) prime \(\geq 5)\), gives rise to the cubic \(E_{a^p, b^p, c^p}: y^2= x(x-a^p) (x-b^p)\), which is a semistable elliptic curve such that the Galois representation of \(\operatorname{Aut}(\overline{\mathbb Q})\) into the Tate module of \(E_{a^p,b^p,c^p}\) at \(p\) is unramified outside of \(2p\). In section 6 Serre’s conjecture on level reduction for modular Galois representations is stated. The next section describes the Mazur-Ribet theorem, which proves a special version of this conjecture and shows that FLT is a consequence of Wiles’ theorem. Furthermore, some other applications are discussed concerning Dénes’ conjecture and the equation \(x^p+y^p +2z^p=0\). The last section presents the famous conjectures of Szpiro and \(abc\), and describes some of their consequences.

The last part of the book is an appendix which gives an historical description of the results, showing that the study of the nontrivial solutions of Fermat’s equation lead to the study of the torsion of the elliptic curve \(E_{A,B,C}\) and conversely.

Reviewer: Dimitros Poulakis (Thessaloniki)

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11D41 | Higher degree equations; Fermat’s equation |

11F11 | Holomorphic modular forms of integral weight |

11G05 | Elliptic curves over global fields |

11G18 | Arithmetic aspects of modular and Shimura varieties |