% This is a revised version of the article ``Introduction to Fermat's % Last Theorem'' by David A. Cox (Amherst College) July 25, 1993 \documentstyle[12pt]{article} \setlength{\textheight}{9in} \addtolength{\textwidth}{1in} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \addtolength{\topmargin}{-.85in} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{definition}{Definition} \newtheorem{corollary}{Corollary} \newtheorem{conjecture}{Conjecture} \newcommand{\bpf} {\noindent {\bf Proof.}} \title{Introduction to Fermat's Last Theorem} \author{David Cox (Amherst College)} \begin{document} \date{July 23, 1993} \maketitle \noindent This article is an introduction to the mathematical history of Fermat's Last Theorem (which we will abbreviate throughout as FLT), broken up into the following periods: \begin{enumerate} \item Diophantus to Euler (250--1783 A.D.) \item Euler to Frey (1783--1982 A.D.) \item Frey to Wiles (1982--1993 A.D.) \end{enumerate} The text that follows is an expanded and corrected version of a lecture given to the Regional Geometry Institute at Smith College on July 13, 1993. The audience included high school teachers, undergraduates, graduate students and researchers in discrete and computational geometry. The mathematical prerequisites are modest---an acquaintance with complex numbers and congruence notation ($a \equiv b \bmod m$ means that $a$ and $b$ are integers which differ by a multiple of $m$) is sufficient for most of what I have to say. Many of the more technical terms are not defined completely and the few proofs that appear are not complete. This article should be regarded as only an introduction to the story of Fermat's Last Theorem. The account given here is neither complete nor definitive. On the other hand, I hope that it succeeds in conveying the flavor of this truly wonderful mathematics. I would like to thank Thomas Colthurst for transcribing the lecture, and I am grateful to my colleagues who pointed out errors in earlier versions of the manuscript. \section{Diophantus To Euler} Our history of FLT starts in 250 A.D. with Diophantus, whose {\em Arithmetica} considered many problems in elementary number theory. A typical problem, taken from Book II, would be to divide a given square into two squares. His solution is as follows: Let the given square be 16, let $x^2$ be one of the required squares, and $(2x-4)^2$ the other square. Therefore, we must satisfy $$ x^2 + (2x-4)^2 = 16 \Rightarrow x^2 + 4x^2 - 16x + 16 = 16 \Rightarrow 5x^2 = 16 x \Rightarrow x = 16/5 $$ so the required squares are $\frac{256}{25}$ and $\frac{144}{25}$. We can observe two things about this solution. First, solutions are presumed to be rational. We neither restrict to only integer solutions nor generalize to real solutions. Second, we care only about finding one solution to a given problem; if we find one, we are happy and move on. The {\em Arithmetica} was one of the last Greek mathematical works translated into Latin; this occured in 1575. Fermat (1601--1665) had a copy of Bachet's translation of 1621 and made a series of intriguing annotations in its margins. Sometime in the late 1630's, while reading the section which solves the problem given above, he added the famous words in the margin: \begin{quote} ``On the other hand, it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvellous proof of this, which however the margin is not large enough to contain.'' \end{quote} Thus the basic claim of Fermat's Last Theorem is that the equation $x^n + y^n = z^n$ has no solutions when $x,y,z$ are nonzero integers and $n > 2$. Generations of mathematical historians have debated over whether Fermat really did have a proof, though many experts doubt that he did. For one thing, the equation $x^n + y^n = z^n$ was atypical for Fermat---the vast majority of the other equations he studied dealt with exponents $\leq 4$. Also, in his correspondence, he only stated FLT for the exponent $n=3$. As for Fermat's ``marvellous proof'', it probably used his technique of infinite descent. His descent proof for $n=4$ is actually known: it follows from a theorem of Fermat's on the area of a right triangle with integral sides not being able to be a square. This proof is given in one of his marginal notes---this time the margin was big enough (though I should point out that FLT wasn't the only instance where it wasn't). It seems likely that Fermat thought that his proofs for $n = 3$ and 4 generalized, and that they almost certainly didn't. So, what happened after Fermat? In 1670, his marginal notes were published by his son. In 1729, Goldbach wrote Euler and mentioned the conjectures of Fermat presented in those notes. This got Euler, only 22 at the time, thinking about number theory. Three years later, Euler wrote his first paper on number theory, disproving a conjecture of Fermat's on primes of the form $2^{2^n} + 1$. For the next fifty years, Euler proved many of Fermat's conjectures and in so doing, transformed number theory from a collection of miscellaneous facts and results into an organized field at the very center of mathematics. Here is an example of what Euler did. In Book VI of the {\em Arithmetica}, Fermat had written in the margin, ``Can one find in whole numbers a square different from 25, when increased by 2, becomes a cube? ... [The answer involves] the doctrine of whole numbers, which is assuredly very beautiful and very subtle ...'' In modern terms, Fermat is claiming that the only integer solutions to $x^3 = y^2 + 2$ are given by $(x,y) = (3, \pm 5)$. You can see the different emphasis from that in Diophantus---Fermat is looking for {\em all} solutions, and he recognizes that asking for integer solutions (rather than rational ones) is a question of independent interest. To prove this, Euler uses numbers of the form $a + b \sqrt{-2}$, with $a,b$ integers. Here is his proof: $$x^3 = y^2 + 2 = ( y + \sqrt{-2} ) ( y - \sqrt{-2})$$ One can show that $y + \sqrt{-2}$ and $y - \sqrt{-2}$ are relatively prime, and since their product is a cube, each of them must also be a cube, so $$ y + \sqrt{-2} = ( p + q \sqrt{-2} )^3 = p^3 - 6pq^2 + (3 p^3 q - 2 q^3 ) \sqrt{-2} $$ $$ \Rightarrow 1 = 3p^2 q - 2 q^3 = q ( 3p^2 - 2q^2 ) $$ The last equation implies $p = \pm 1$ and $q = 1$. Plugging this in, we get $y = p^3 - 6 pq^2 = \pm 5 $ and $ x = 3$, as claimed. This proof, while elegant, is incomplete, for we do not know that numbers of the form $a + b \sqrt{-2}$ have {\em unique factorization}, or even for that matter, {\em primes} (athough it is relatively easy to prove that the numbers $a+b\sqrt{-2}$ have these properties). There are three reasons why the above example is important: \begin{itemize} \item First, it reminds us that there are lots of diophantine equations besides just FLT, and that what we really want is a method for dealing with as many of them as possible. \item Second, it generalizes the integers to a set of numbers which has much of the same arithmetic structure (addition, multiplication, etc.) This sort of generalization occurs frequently in mathematics. \item Finally, the equation $y^2 = x^3 -2$ is an example of an {\em elliptic curve}. Elliptic curves will play a crucial role in the final proof of FLT. \end{itemize} \section{Euler to Frey} This section is only a sketch of more than two hundred years of beautiful and wonderful number theory. For more information on the work on FLT done during this period, we warmly recommend both Edwards' {\sl Fermat's Last Theorem} (Springer, 1977) and Ribenboim's {\sl 13 Lectures on Fermat's Last Theorem} (Springer, 1979). Before we begin, let us first observe that it suffices to prove FLT for $n=4$ (done by Fermat) and for $n$ an odd prime (since we can factor the exponent). We can also assume that $x,y,z$ are nonzero relatively prime integers (because we can cancel common factors). That being said, here are some of the highlights of the 19th century work on FLT: \begin{itemize} \item By the early 1800's, all of Fermat's problems were solved except for FLT (thus justifying the name, Fermat's Last Theorem). \item 1816 --- The French Academy announces a prize for solutions to FLT. \item In the 1820's, Sophie Germain shows that if $p$ and $2p+1$ are prime, then $x^p + y^p = z^p$ has no solution with $p\! \not\,\mid x y z$. This is the so-called Case I of FLT. [Case II is where $p \mid x y z$ and is usually regarded as being much harder.] \item 1825 --- Dirichlet and Legendre prove FLT for $n = 5$. \item 1832 --- Dirichlet, after trying to prove it for $n=7$, proves FLT for $n = 14$. \item 1839 --- Lam\'{e} proves FLT for $n=7$. \item 1847 --- Lam\'{e} and Cauchy present false proofs of FLT \item 1844--1847 --- Kummer's work on FLT \end{itemize} We will describe Kummer's extremely important work on FLT in more detail. Kummer (and Cauchy and Lam\'e) started, \`a la Euler, by factoring the right hand side of the FLT equation as $$ x^p = z^p - y^p = (z - y)(z - \zeta y)(z - \zeta^2 y) \ldots (z - \zeta^{p-1} y ) $$ where $\zeta = e^{2 \pi i/p} = \cos( 2 \pi / p ) + i \sin(2 \pi / p ) $ is a {\em $p^{th}$ root of unity} and satisfies $\zeta^p = 1$. In general, working with roots of unity will require us to use numbers of the form $$ a_0 + a_1 \zeta + \ldots + a_{p-1} \zeta^{p-1},\quad a_0 \ldots a_{p-1} \in {\bf Z} $$ which are called {\em cyclotomic integers}. But a problem arises when unique factorization, one of our main tools, fails for the cyclotomic integers. As Kummer discovered in 1844, this first occurs for $p = 23$ (and now we know that unique factorization fails for all bigger primes as well). Kummer's solution to this was twofold. First, he introduced a generalization of cyclotomic integers, called {\em ideal numbers}, which make up for the lack of unique factorization. Second, he defined the {\em class number $h$}, which measures how badly unique factorization fails. Here is a summary of Kummer's results: \begin{itemize} \item 1847 --- Theorem: FLT holds for $p$ if $p\! \not\,\mid h$ (such $p$ are called {\em regular primes}). \item 1847 --- Theorem: $p$ is regular iff $p$ doesn't divide the numerator of the Bernoulli numbers $B_2, B_4, \ldots, B_{p-3}$. We can define the Bernoulli numbers by $$ \frac{x}{e^x-1} = \sum_{n=1}^{\infty} \frac{B_n}{n!} x^n $$ A corollary of this result is that for $p < 100$, only 37, 59, and 67 are irregular. \item 1850 --- The French Academy offers a second prize for a solution to FLT, withdraws it, and then awards a medal to Kummer. \item 1857 --- Kummer develops complicated criteria for proving FLT for certain irregular primes. There are some gaps in his proofs which are later filled in by Vandiver in the 1920's. These results establish FLT for $p < 100$. \end{itemize} The above history makes a wonderful story about how FLT inspired one of the greatest inventions in number theory, but the story is unfortunately false. Kummer was actually not trying to prove FLT, but something called a reciprocity theorem. Reciprocity theorems have their origins in Fermat's study of equations like $p = x^2 + y^2$ and $p = x^2 + 2y^2$. In trying to understand these results, Euler, Lagrange, Legendre and Gauss created the theory of quadratic forms and proved the law of quadratic reciprocity. Later, Gauss, Abel and Jacobi formulated versions of cubic and biquadratic reciprocity, and Kummer and Eisenstein made the first attempts at higher reciprocity laws. Cyclotomic integers and ideal numbers came about primarily from Kummer's attempts to prove these higher reciprocity laws. In turn, these concepts not only had something interesting to say about FLT, but they also made significant contributions toward the development of class field theory and abstract algebra (we use the terminology ``ideal of a ring'' because of Kummer's ``ideal numbers''). Here are some highlights of the history of FLT after Kummer: \begin{itemize} \item 1908 --- The Wolfskehl prize for a solution to FLT is announced. Later inflation in the Deutschmark reduces the value of this prize considerably, but does not reduce the flow of crank solutions submitted. \item 1909 --- Wieferich proves if $x^p + y^p = z^p$ and $p\! \not\,\mid xyz$ (Case I of FLT), then $2^{p-1} \equiv 1 \bmod {p^2}$. This is a strong congruence which is particularly easy to check on a computer. \item 1953 --- Inkeri proves that if $x^p + y^p = z^p$ and $x < y < z$, then $x > ((2p^3+p)/\log(3p))^p$ in Case I and $x > p^{3p-4}$ in Case II. \item 1971 --- Brillhart, Tonascia, and Weinberger show that Case I of FLT is true for all primes less than $3 \cdot 10^9$. \item 1976 --- Wagstaff shows that FLT is true for all primes less than 125,000. \end{itemize} The conclusion of all this work is that any counterexample to FLT must involve $p \geq 125,003$ and $z > y > x > (125,003)^{375,005} \approx 4.5\cdot10^{1,911,370}$. (We should also mention that by 1992 the lower bound on the exponent had been raised to $p > 4,000,000$.) \section{Frey to Wiles} In 1983, Faltings proved the Mordell Conjecture, which implies that a polynomial equation with rational coefficients $Q(x,y) = 0$ has only finitely many rational solutions when the curve has genus $\ge 2$ (the genus is defined in terms of the complex solutions of the equation). Since $x^n + y^n = 1$ has genus $\geq 2$ for $n \geq 4$, the Mordell Conjecture implies there are only finitely many rational solutions. Then, clearing denominators, it follows easily that $x^n +y^n = z^n$ has only finitely many relatively prime integer solutions. In one sense, this is not so useful, since we want to show that the number of solutions is actually zero. But Filaseta, Granville and Heath-Brown were able to use this result to show that FLT holds for ``most'' exponents (in the sense that if you look at exponents from 3 to $N$, the percentage where FLT could fail becomes smaller and smaller as $N$ increases). Also, the Mordell Conjecture is an extremely strong statement about an entire class of equations, not just FLT, and in proving the Mordell Conjecture, Faltings used the modern machinery of algebraic geometry, which had been developing since the 1950's. The really interesting part of the story begins with Frey's work from 1982 to 1986. Frey showed that nontrivial solutions to FLT give rise to very special elliptic curves, which we shall call Frey curves. The importance of Frey curves is indicated by the fact that elliptic curves are a large and important part of modern number theory, and more importantly, a number of standard conjectures in number theory imply that Frey curves can't exist. If $a^p + b^p = c^p$ is a solution to FLT, then the associated Frey curve is $$y^2 = x ( x - a^p ) ( x + b^p ) $$ As usual, we are assuming that $a,b,c$ are nonzero relatively prime integers and $p$ is an odd prime. This is an elliptic curve over the rational numbers ${\bf Q}$, similar to the $y^2 = x^3 - 2$ considered by Fermat. In general, an elliptic curve over ${\bf Q}$ is given by an equation of the form $$ y^2 = a x^3 + b x^2 + c x + d $$ with $a,b,c,d$ rational and the cubic polynomial in $x$ on the right hand side of the equation having distinct roots. Actually, we have to be bit careful when constructing the Frey curve. A solution $a^p+b^p = c^p$ gives rise to solutions $b^p+a^p = c^p$ and $a^p + (-c)^p = (-b)^p$ (since $p$ is odd). From here it is easy to rearrange the solution so that $b$ is even and $a \equiv -1 \bmod 4$. This is needed in order that the Frey curve be {\em semistable} (this concept will be discussed below). We will also assume that $p > 3$. By the late 1980's, there are three ways in which Frey curves, combined with standard conjectures, could prove FLT: \begin{itemize} \item The Bogomolov--Miyaoka--Yau (BMY) inequality for arithmetic surfaces relates various invariants of a curve defined over the integers. This inequality is an arithmetic analog of a well known inequality for complex surfaces. By a theorem of Parshin's, this inequality implies the Szpiro Conjecture, which relates the minimal discriminant to the conductor of an elliptic curve. The discriminant and conductor are two invariants of elliptic curves which we will define later. It was known that the Szpiro Conjecture implies FLT for all large primes $p$. \item A conjecture of Vojta concerns heights of points (relative to the canonical class) of a curve defined over the integers. This conjecture implies the Mordell Conjecture, and it also implies FLT for large exponents $n$. \item The Taniyama--Shimura Conjecture (which states that all elliptic curves over the rational numbers are modular---we will give a more precise statement below), together with a conjecture of Serre on level reduction for modular Galois representations, imply FLT for {\em all} $p$. \end{itemize} In 1988, Miyaoka (the M in BMY) gave a lecture in Bonn in which he stated the arithmetic BMY inequality as a theorem, thus proving FLT (for large $p$) by our first route. In the days following his lecture, there was much fanfare in the press, so it was rather disappointing when a week later he had to retract his proof because an error had been found in the argument. The conjecture of Vojta is still open and is representative of a whole class of questions and conjectures that have been made concerning the size and location of rational solutions of certain equations with integer solutions. More on this subject can be found in Lang's book {\sl Number Theory III: Diophantine Geometry} (Springer, 1991). In particular, see pages 63--64 for a discussion of Vojta's conjecture and FLT. The story of the third route to FLT is the one that concerns us here. In 1985 Frey tried to prove that Taniyama--Shimura implies FLT using a certain kind of level reduction, but there were serious gaps in his proof. Several people tried to fix Frey's argument, and it was Serre who saw that a conjecture on level reduction for modular Galois representations would fill the gap. Hence we may credit Frey and Serre with showing that FLT follows from Taniyama--Shimura and the level reduction conjecture made by Serre. Then, in 1986, Ribet made significant progress along this route to FLT by proving Serre's conjecture. Thus FLT was now a consequence of the Taniyama--Shimura Conjecture. Inspired by this development, Andrew Wiles began to work on the Taniyama--Shimura Conjecture, and seven years later, he presented a proof on June 23, 1993 that it is true for semistable elliptic curves, which (as we will see below) is good enough to prove FLT. Wiles' proof is reportedly in a two hundred page manuscript, which has not yet been released. But many people in the mathematical community are confident that the proof will hold up under careful scrutiny. For a broad outline of Wiles' argument, see the article by Ken Ribet in the forthcoming issue of the {\sl Notices of the AMS} (Ribet's article also has some useful references). One interesting observation is that Frey was not the first to link FLT to elliptic curves. Previous connections had been made, often with the goal of using known facts about FLT to prove theorems about elliptic curves. However, on page 262 of {\it Points d'ordre $2p^h$ sur les courbes elliptiques} (Acta Arith.~{\bf 26} (1975), 253--263), Hellegouarch writes down the Frey curve for a solution to FLT of exponent $2p^h$. But Frey was clearly the first to suspect that the Frey curve could not exist because of the Taniyama--Shimura Conjecture. To explain the Taniyama--Shimura Conjecture, we must first define modular functions. \begin{definition} A function $f(z)$ on the upper half plane $\{ x + i y : y > 0 \}$ is {\em modular of level $N$} if \begin{enumerate} \item $f(z)$ is meromorphic (even at cusps) [this is the analog of being differentiable for complex functions] \item For any matrix $\Big(\! \begin{array}{cc} a & b \\ c & d \end{array}\! \Big)$ with $ad - bc = 1$, $a,b,c,d $ integers and $N \mid c$, we have $$f\Big( \frac{ az + b }{ c z + d } \Big) = f(z) $$ \end{enumerate} \end{definition} \begin{conjecture}[Taniyama--Shimura] Given an elliptic curve $y^2 = ax^3 + bx^2 + cx + d$ over ${\bf Q}$, there are nonconstant modular functions $f(z), g(z)$ of the same level $N$ such that $$f(z)^2 = ag(z)^3 + bg(z)^2 + cg(z) + d$$ \end{conjecture} Thus the Taniyama--Shimura Conjecture says that elliptic curves over ${\bf Q}$ can be parameterized by modular functions. Such an elliptic curve is said to be {\em modular}. Wiles proved this conjecture for semistable elliptic curves. We should mention that our statement of the conjecture is very naive and in fact is not even complete---one also needs to require that the parametrization be ``defined over ${\bf Q}$'' in a suitable sense. In practice, mathematicians work with more sophisticated definitions of what it means for an elliptic curve to be modular. See pages 130--135 of Lang's book (cited above) for a more complete discussion of the conjecture (which includes some of its history). Besides modular functions, we also need to know about modular forms of weight 2. The easiest way to see how these arise is through elliptic integrals. An elliptic integral is an integral of the form $$\int \frac{dx}{\sqrt{ax^3 + bx^2 + cx+ d}} $$ [Strictly speaking, this is only an elliptic integral of the first kind---there are many other types of elliptic integrals.] If $y^2 = ax^3 + bx^2 + cx + d$, then this integral is simply $\int \frac{dx}{y} $. If our curve is modular, then $x = f(z)$, $y = g(z)$, and $$\frac{dx}{y} = \frac{df}{g} = \frac{f'(z) dz}{g(z)} = F(z) dz$$ Because of the way $F(z)$ transforms under the matrices in Definition 1, we call $F(z)$ {\em a modular form of weight 2 and level $N$}. The function $F(z)$ has some remarkable properties. It is holomorphic and vanishes at the cusps, and for this reason is called a {\em cusp form}. In addition, $F(z)$ is an {\em eigen-form} for the action of a certain Hecke algebra on the space of all cusp forms. So $F(z)$ is a rather sophisticated object. The miracle is that $F(z)$ is intimately connected to the curve $y^2 = ax^3 + bx^2 + cx + d$. Roughly speaking, one can reconstruct $F(z)$ simply by knowing the number of solutions of the congruences $y^2 \equiv ax^3+bx^2+cx+d \bmod p$ for all primes $p$. Then the fact that $F(z)$ is a cusp form of weight 2 and level $N$ tells us some profound things about the elliptic curve. This is one of the reasons why Taniyama--Shimura is such a wonderful conjecture---number theorists would be excited by its proof even if there were no connection to FLT. We can now sketch the argument of Frey and Serre that shows why FLT follows from Taniyama--Shimura and the level reduction conjecture of Serre. We begin with the FLT solution $a^p + b^p = c^p$. As above, we will assume that $p > 3$ is prime and that $a,b,c$ are relatively prime with $b$ even and $a \equiv -1 \bmod 4$. The first step is to compute some invariants of the Frey curve $y^2 = x(x-a^p)(x+b^p)$: \begin{itemize} \item The {\em discriminant} of the cubic polynomial $x(x-a^p)(x+b^p)$ is the product of the squares of the differences of the roots $$( a^p - 0 )^2 ( - b^p - 0 )^2 ( a^p - (-b^p))^2$$ By our assumption that $a,b,c$ is a solution to FLT, this equals to $a^{2p}b^{2p}c^{2p}$. \item Besides the discriminant just defined, an elliptic curve has a more subtle invariant called the {\it minimal discriminant}. One can show that the minimal discriminant of the Frey curve is $2^{-8}a^{2p}b^{2p}c^{2p}$. Since $b$ is even and $p \ge 5$, this is still an integer. [The difference is that the discriminant depends on the particular equation defining the curve, while the minimal discriminant is intrinsic to the curve itself.] \item The {\em conductor} of the Frey curve is $N = \prod_{p \mid abc} p$. A more precise form of the Taniyama--Shimura Conjecture asserts that the conductor equals the level $N$ of the modular functions that parametrize the curve. \item The {\em $j$-invariant} of the Frey curve is $j = 2^8 \frac{(a^{2p} + b^{2p} + a^pb^p )^3}{a^{2p}b^{2p}c^{2p}}$. \end{itemize} We then have the following results about the Frey curve: \begin{lemma} The Frey curve is semistable. \end{lemma} \bpf\ We first need to define what semistable means. When a prime $l$ divides the discriminant, two or possibly all three of the roots become congruent mod $l$. Roughly speaking, an elliptic curve is semistable if for all such primes $l$, only two roots become congruent mod $l$ (the definition is a bit more complicated when dealing with the primes 2 and 3). Thus, for primes bigger than 3, the Frey curve is semistable since the discriminant is $a^{2p}b^{2p}c^{2p}$ and the roots are 0, $a^p$ and $-b^p$, where $a^p$ and $b^p$ are relatively prime. More work is required to check semistability at $l = 2$ or 3, and when $l = 2$, we use the conditions $b$ even, $a \equiv -1 \bmod 4$ and $p > 3$. Q.E.D. \begin{corollary}[Wiles] The Frey curve is modular. \end{corollary} \begin{lemma} For every odd prime $l$ dividing $N$, the $j$-invariant of the Frey curve can be written as $l^{-mp} \cdot q$, where $m$ is a positive integer and $q$ is a fraction not involving $l$. [We say that $j$-invariant is {\em exactly divisible} by $l^{-mp}$ in this case.] \end{lemma} \bpf\ The power of $l$ dividing the denominator of the $j$-invariant is obviously a multiple of $p$. The numerator of the $j$-invariant is $2^8(a^{2p} + b^{2p} + a^pb^p)^3 = 2^8(c^{2p} - b^pc^p)^3$. We know that $l$ divides one of $a,b,c$ because $l \mid N$. Since $a,b,c$ are relatively prime and $l$ is odd, it follows that $l$ can't divide the numerator, and the lemma is proved. The lemma fails for $l = 2$ because of the factor of $2^8$ in numerator. Q.E.D. \medskip In the context of these three results---semistable modular elliptic curves whose $j$-invariants are exactly divisible by $l^{-{\rm multiple\ of}\ p}$ for odd primes $l$ dividing $N$---the level reduction conjecture of Serre (to be discussed below) now applies for {\em all} odd primes dividing $N$. We can now prove Fermat's Last Theorem: \begin{theorem} The equation $x^p + y^p = z^p$ has no solutions with $a,b,c$ nonzero for $p$ an odd prime. \end{theorem} \bpf\ Suppose there were a solution $a^p + b^p = c^p$, with our usual assumptions about $p$ and $a,b,c$. Then we have a Frey curve, which by Corollary 1 has a cusp form $F$ of weight 2 and level $N$. This curve also has a Galois representation $\rho$ on the points of order $p$ on the curve (we won't be able to define precisely what this means). The form $F$ is linked to the representation $\rho$ in an especially nice way. As we observed above, the hypotheses of Serre's level reduction conjecture are satisfied for all odd primes $l$ dividing $N$. In such a case, this conjecture (proved by Ribet) asserts that there is a cusp form $F'$ of weight 2 and level $N/l$ with $$ F' \equiv F \bmod {p}$$ and $F'$ is also an eigen-form for the appropriate Hecke algebra (it takes some work to define what it means for modular forms to be congruent modulo $p$). This congruence means that $F'$ is linked to $\rho$ in the same way $F$ was, except that $F'$ has smaller level $N/l$. But then, if $l'$ is another odd prime dividing $N$, we can apply the level reduction conjecture again to $F'$ and get a cusp form $F''$ with even smaller level $N/l l'$, and then apply it again to $F''$, etc. Eventually we get a cusp form $\tilde{F}$ of weight 2 and level 2. [Note that $2$ divides $N$ since $b$ is even.] Here is a diagram of the argument so far: \[ \begin{array}{rcccl} \hbox{solution of FLT} & \to & \hbox{Frey curve} & \to & \hbox{cusp form of level $N$} \\ &&& \to & \hbox{cusp form of level $N/l$}\\ &&& \to & \hbox{cusp form of level $N/ll'$}\\ &&& \vdots & \\ &&& \to & \hbox{cusp form of level $2$} \end{array} \] But it is well known that there are no cusp forms of weight 2 and level 2. Hence the above diagram self-destructs, and Fermat's Last Theorem is proved! Q.E.D. \medskip This brings us to the end of the article, but certainly not to the end of the story. One thing missing from this account of Fermat's Last Theorem is the work of the many mathematicians who created the theories of elliptic curves, modular forms and Galois representations, and searched out the amazing connections between them. There is a lot more to say about the mathematics involved in the proof of Fermat's Last Theorem! \bigskip \hskip 4truein David A. Cox \hskip 4truein Department of Mathematics \hskip 4.2truein and Computer Science \hskip 4truein Amherst College \hskip 4truein Amherst, MA 01002 \hskip 4truein dac@cs.amherst.edu \end{document}