 # Prova do último teorema de Fermat - A. Wiles

(Parte 1 de 7)

i Nigel Boston

University of Wisconsin - Madison i INTRODUCTION.

This book will describe the recent proof of Fermat’s Last Theorem by Andrew Wiles, aided by Richard Taylor, for graduate students and faculty with a reasonably broad background in algebra. It is hard to give precise prerequisites but a first course in graduate algebra, covering basic groups, rings, and fields together with a passing acquaintance with number rings and varieties should suffice. Algebraic number theory (or arithmetical geometry, as the subject is more commonly called these days) has the habit of taking last year’s major result and making it background taken for granted in this year’s work. Peeling back the layers can lead to a maze of results stretching back over the decades.

I attended Wiles’ three groundbreaking lectures, in June 1993, at the Isaac Newton Institute in Cambridge, UK. After returning to the US, I attempted to give a seminar on the proof to interested students and faculty at the University of Illinois, Urbana-Champaign. Endeavoring to be complete required several lectures early on regarding the existence of a model over

Q for the modular curve X0(N) with good reduction at primes not dividing N. This work hinged on earlier work of Zariski from the 1950’s. The audience, keen to learn new material, did not appreciate lingering over such details and dwindled rapidly in numbers.

Since then, I have taught the proof in two courses at UIUC, a two-week summer workshop at UIUC (with the help of Chris Skinner of the University of Michigan), and most recently a course in spring 2003 at the University of Wisconsin - Madison. To avoid getting bogged down as in the above seminar, it is necessary to assume some background. In these cases, references will be provided so that the interested students can fill in details for themselves. The aim of this work is to convey the strong and simple line of logic on which the proof rests. It is certainly well within the ability of most graduate students to appreciate the way the building blocks of the proof go together to give the result, even though those blocks may themselves be hard to penetrate. If anything, this book should serve as an inspiration for students to see why the tools of modern arithmetical geometry are valuable and to seek to learn more about them.

An interested reader wanting a simple overview of the proof should consult Gouvea , Ribet , Rubin and Silverberg , or my article . A much more detailed overview of the proof is the one given by Darmon, Diamond, and Taylor , and the Boston conference volume  contains much useful elaboration on ideas used in the proof. The Seminaire Bourbaki article by Oesterle and Serre  is also very enlightening. Of course, one should not overlook the original proof itself ,  .

iv CONTENTS.

Introduction.

Contents.

Chapter 1: History and overview.

Chapter 2: Profinite groups, complete local rings.

Chapter 3: Infinite Galois groups, internal structure.

Chapter 4: Galois representations from elliptic curves, modular forms, group schemes.

Chapter 5: Invariants of Galois representations, semistable representations.

Chapter 6: Deformations of Galois representations.

Chapter 7: Introduction to Galois cohomology.

Chapter 8: Criteria for ring isomorphisms.

Chapter 9: The universal modular lift.

Chapter 10: The minimal case.

Chapter 1: The general case. Chapter 12: Putting it together, the final trick.

1 History and Overview

It is well-known that there are many solutions in integers to x2+ y2 = z2, for instance (3,4,5),(5,12,13). The Babylonians were aware of the solution (4961,6480,8161) as early as around 1500 B.C. Around 1637, Pierre de Fermat wrote a note in the margin of his copy of Diophantus’ Arithmetica stating that xn+yn = zn has no solutions in positive integers if n > 2. We will denote this statement for n (FLT)n. He claimed to have a remarkable proof. There is some doubt about this for various reasons. First, this remark was published without his consent, in fact by his son after his death. Second, in his later correspondence, Fermat discusses the cases n = 3,4 with no reference to this purported proof. It seems likely then that this was an off-the-cuff comment that Fermat simply omitted to erase. Of course (FLT)n implies (FLT)αn, for α any positive integer, and so it suffices to prove (FLT)4 and (FLT) for each prime number > 2.

1.1 Proof of (FLT)4 by Fermat vi 1.1 Proof of (FLT)4 by Fermat

First, we must deal with the equation x2 + y2 = z2. We may assume x, y, and z are positive and relatively prime (since otherwise we may divide out any common factors because the equation is homogeneous), and we see that one of x or y is even (since otherwise z2 ≡ 2 (mod 4), which is a contradiction). Suppose that x is even. Then(z − y 2 with relatively prime factors on the left hand side and a square on the right hand side. Hence which yields the same result.]

Theorem 1.1 x4 + y4 = z2 has no solutions with x,y,z all nonzero, relatively prime integers.

with a and b relatively prime. Clearly, b is even (y is odd, since x is even), and from a2 = y2 + b2 we get

Note, however, that z > a2 = (g2)2 > g, and so we are done by infinite descent (repeated application produces an infinite sequence of solutions with ever smaller positive integer z, a contradiction). QED

The first complete proof of this case was given by Karl Gauss. Leonhard Euler’s proof from 1753 was quite different and at one stage depends on a fact that Euler did not justify (though it would have been within his knowledge to do so). We outline the proof - details may be found in , p. 285, or , p. 43.

Gauss’s proof leads to a strategy that succeeds for certain other values of n too. We work in the ring A = Z[ζ] = {a+bζ : a,b ∈ Z}, where ζ is a primitive cube root of unity. The key fact here is that A is a PID and hence a UFD. We also repeatedly use the fact that the units of A are precisely ±ζi (i = 0,1,2).

Proof: By homogeneity, we may assume that x,y,z are relatively prime. Factoring x3 + y3 = uz3 gives where the gcd of any 2 factors on the left divides λ := 1 − ζ. If each gcd is 1, then each factor is a cube up to a unit. In any case, λ is “small” in that |A/(λ)| = 3. In particular, each element of A is either 0,±1 mod λ.

Lemma 1.3 There are no solutions when λ 6 |xyz.

Proof: In this case, the gcd of any 2 factors on the left is λ. Hence we can assume that

1.3 Further Efforts at Proof x 1.3 Further Efforts at Proof

Peter Dirichlet and Adrien Legendre proved (FLT)5 around

 Z[ζ] = {a0 + a1ζ + + al−2ζl−2 : ai ∈ Z},

then there are cases when Z[ζ] is not a UFD and the factorization method used above fails. (In fact, Z[ζ] is a UFD if and only if ≤ 19.)

It turns out that the method can be resuscitated under weaker conditions. In 1844 Ernst Kummer began studying the ideal class group of Q(ζ), which is a finite group that measures how far Z[ζ] is from being a UFD . Between 1847 and 1853, he published some masterful papers, which established almost the best possible result along these lines and were only really bettered by the recent approach detailed below, which began over 100 years later. In these papers, Kummer defined regular primes and proved the following theorem, where h(Q(ζ)) denotes the order of the ideal class group.

Remark 1.7 The first irregular prime is 37 and there are infinitely many irregular primes. It is not known if there are infinitely many regular primes, but conjecturally this is so.

Theorem 1.8 (Kummer) (i) (FLT) holds if is regular. (2) is regular if and only if does not divide the numerator

1.3 Further Efforts at Proof xi

Here Bn are the Bernoulli numbers defined by x

For instance, the fact that B12 = − 6912730 shows that 691 is irregular. We shall see the number 691 appearing in many dif- ferent places.

Here the study of FLT is divided into two cases. The first case involves showing that there is no solution with 6 |xyz. The idea is to factor x + y = z as

where ζ = e2pii/. The ideals generated by the factors on the left side are pairwise relatively prime by the assumption that 6 |xyz (since λ := 1 − ζ has norm - compare the proof of

(FLT)3), whence each factor generates an th power in the ideal class group of Q(ζ). The regularity assumption then shows that these factors are principal ideals. We also use that any for unit u in Z[ζ], ζsu is real for some s ∈ Z. See  or  for more details.

The second case involves showing that there is no solution to

FLT for |xyz. In 1823, Sophie Germain found a simple proof that if is a prime with 2 + 1 a prime then the first case of (FLT)` holds. Arthur Wieferich proved in 1909 that if is a prime with

(Parte 1 de 7)