From: radcliff@csd4.csd.uwm.edu (David G Radcliffe) Newsgroups: sci.math Subject: Fermat's Last Theorem Talk Date: 17 Jul 1993 06:59:37 GMT This is an article I found in the Geometry Forum. I thought that many readers of sci.math would find it interesting. At least, it should be a pleasant diversion from the interminable debate about the Polly Nomial story. ------------------------------------------------------------------ From: thomasc@geom.umn.edu (thomasc) Organization: Geometry Center, University of Minnesota This is the transcription of a talk given by David Cox on Fermat's Last Theorem, here at the Smith College RGI on July 13. Don't ask me what FLT has to do with geometry. -Thomas C P.S. 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A typical problem, taken from Book II, would be to divide a given square into two squares. His solution was 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 $$ x^2 + (2x-4)^2 = 16 \implies x^2 + 4x^2 - 16x + 16 = 16 \implies 5x^2 = 16 x \implies x = 16/5 $$ 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. Later, we will look at the problems involved in finding all solutions to an equation. The {\em Arithmetica} was one of the last Greek mathematical works translated into Latin; this occurred in 1575. Fermat (1601-1665) had a copy of Bachet's translation of 1621, and made extensive 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: \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} Generations of mathematical historians have debated over whether Fermat really did have a proof, and the consensus of the experts is that he did not. Let us present their evidence. First, $x^n + y^n = z^n$ was a very atypical equation for Fermat. All the other equations he studied dealt with exponents $\leq 4$. Also, he only stated FLT in his letters for the case of $n=3$, and his proof of this probably used infinite descent. The proof of the next case $n=4$ is easier and in fact follows from a theorem Fermat had proved on the area of a right triangle with integral sides not being able to be a square. This proof, in fact, was given in one of his marginal notes; sometimes the margin was big enough, and sometimes (as in FLT and several other instances) 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, based on a conjecture of Fermat's on primes of the form $2^{2^n} + 1$. And the rest, as they say, is history. For the next forty 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 $y^3 = x^2 + 2$ 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 used numbers of the form $a + b \sqrt{-2}$, with $a,b$ integers. Here is his proof: $$x^3 = y^2 + 2 = ( y + \sqrt{-2} ) ( y - \sqrt{-2} ) $$ We 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 $$ y + \sqrt{-2} = ( p + q \sqrt{-3} )^3 = p^3 - 6pq^2 + (3 p^3 q - 2 q^3 ) \sqrt{-2} $$ $$ \implies 1 = 3p^2 q - 2 q^3 = q ( 2p^2 - 2q^2 ) $$ And $p = \pm 1$ and $q = 1$. Plugging this in, we get $y = p^3 - 6 pq^2 = \pm 5 $ and $ x = 3$. 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}. We will come back to these problems in the next section. For the moment, here are three reasons why the above example is important. First, it reminds us that there are a whole 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. 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. 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. \section{Euler to Faltings} This section is but a mere sketch of more than two hundred years of beautiful and wonderful number theory. For more information on this period, we recommend both Edwards's ``Fermat's Last Theorem'' (Springer 1977) and Ribenboim's ``13 Lectures on Fermat's Last Theorem'' (Springer 1979). Before we begin, let us observe that it suffices to prove FLT for $n=4$, which was done by Fermat, and for $n$ an odd prime (since we can factor the exponent) and for $x,y,z$ relatively prime (because we can cancel common factors). That being said, here are some of the highlights of the 19th century: \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, the second case being where $p \mid x y z$. The second case is much harder. \item 1825 --- Dirichlet and Legendre prove FLT for $n = 5$. \item 1832 --- Dirichlet, after trying hard 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} Let us dwell a bit on Kummer's extremely important work on FLT. Kummer started, \'a la Euler, by factoring the right hand side of the FLT equation as $$ x^p = z^p - y^p = (z - y)(z - \zeta y)(z - \zeta^2) \ldots (z - \zeta^{p-1} y ) $$ 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 $$ a_o + a_1 \zeta + \ldots + a_{p-1} \zeta^{p-1}, a_1 \ldots a_{p-1} \in \zz $$ which are called {\em cyclotomic integers}. But a problem arises when unique factorization, one of our main tools, fails for the cyclotomic integers. This first occurs for $p = 23$ (and actually, for all $p \geq 23$), as Kummer discovered in 1844. Kummer's solution to this was twofold. First, he introduced a generalization of the cyclotomic integers, called {\em the 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 history 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 {\em reciprocity theorem}. Reciprocity theorems started with Fermat in his solutions to equations like $p = x^2 + y^2$ and $p = x^2 + 2y^2$, which were later generalized into quadratic reciprocity by Euler, Lagrange, Legendre, and Gauss. 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 towards the development of class field theory and abstract algebra (for example, we today 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 $2^{p-1} \equiv 1 \pmod{p^2}$. 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 $x > p^{3p-4}$. \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 $z > y > x > (125,003)^{375,005} \approx 4.5\cdot10^{1,911,370}$. \section{Faltings to Wiles} In 1983, Faltings proved the Mordell Conjecture, which states that when a polynomial in two variables $Q(x,y)$ has rational coefficients and {\em genus} $\geq 2$, then $Q(x,y) = c$ has only finitely many rational solutions. The genus of a polynomial is simply its topological genus (i.e., number of holes) with the equation considered as a surface over the complex numbers. Since $x^n + y^n = 1$ has genus $\geq 2$ for $n \geq 4$, it follows from the Mordell Conjecture that $x^n +y^n = z^n$ has only finitely many relatively prime solutions. This was a significant development. Mordell's conjecture is an extremely strong statement about an entire class of equations, not just FLT. Also, in proving the Mordell Conjecture, Faltings used the modern machinery of algebraic geometry, which had been developing since WW II. The next breakthrough occured in 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 {\em Frey curves}. The importance of Frey curves is indicated by the fact that a number of standard conjectures in number theory imply that they can't exist. If $a^p + b^p = c^p$ is a solution to FLT, then the associated Frey curve is $$y^2 = x ( x + b^p ) ( x + c^p ) $$ This is an elliptic curve, similar to the $y^2 = x^3 - 2$ considered by Fermat. In general, an elliptic curve over the rational numbers 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. Elliptic curves are a large and important part of modern number theory, and by connecting FLT to them, Frey cleared the way for some powerful machinery to be brought to bear. There are two main ways in which Frey's work, combined with standard conjectures, could prove FLT. The first is based on the conjectured Bogomolv -- Miyaoka -- Yau (BMY) inequality for arithmetic surfaces. By a theorem of Parshin's, the truth of this inequality would imply 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. By Frey's work, Szpiro's conjecture would imply FLT for all large $p$. The second way combines the Taniyama --- Weil conjecture (which states that all elliptic curves over the rational numbers are modular) and a conjecture of Serre on modular Galois representations. Frey proved that if both of these were true, then FLT would follow for {\em all} $p$. In 1988, Miyaoka (the M in BMY) announced a proof of the BMY inequality, and thus (by our first route) a proof of FLT. This announcement was followed by much fanfare in the press, so it was rather disappointing for all involved when he had to retract his proof after the first of two lectures when an error was found. In the period of 1987 -- 1990, Ribet made progress along the second route to FLT by proving the Serre conjecture. Independently in the period from 1986 -- 1993, Andrew Wiles worked on the Taniyama --- Weil conjecture, and eventually presented a proof on June 23 that it is true for semistable elliptic curves, which is good enough to prove FLT. Wiles' proof is reportedly in a two hundred page manuscript, which has not yet been released. But many of the experts who have examined parts of the proof are confident that it will hold up under careful scrutiny. We will now sketch the details of how Wiles' proof implies FLT. To explain what the Taniyama --- Weil conjecture is, 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---Weil] Given an elliptic curve $y^2 = a x^3 + b x^2 + c x + d$, there are nonconstant modular functions $f(z), g(z)$ such that $$f(z)^2 = a g(z)^3 + b g(z)^2 + c g(x) + d$$ \end{conjecture} Another way of looking at the Taniyama---Weil conjecture is that it says that elliptic curves are parameterizable by modular functions. This is what Wiles proved for {\em semistable } (this will be defined later) elliptic curves. Of course, this is a very naive way of stating the conjecture. In practice, mathematicians work with more sophisticated definitions of what it means for an elliptic curve to be modular. Besides modular functions, one also needs to know about {\em modular forms of weight 2}. The easiest way to see how these arise is through elliptic integrals. An elliptic integral is simply 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 $y^2 = x^3 + Ax + B$, then this integral is simply $\int \frac{dx}{y} $. If our curve is modular, then $x = f(z)$, $y = g(z)$, and $$ \frac{dx}{y} = \frac{df}{g} = \frac{f'(x) dz}{g(z)} = F(z) dz$$ 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 $N$}. $F(z)$ turns out to be intimately connected to the curve $y^2 = a x^3 + bx^2 + c x + d$; its properties say a lot about the properties of the elliptic curve. This is one of the reasons why Weil-Taniyama is so powerful. $F(z)$ is also a very nice function in and of itself; it is holomorphic and vanishes at the cusps. For this reason, it is called a {\em cusp form}. Here are some invariants of the Frey curve $y^2 = x(x+b^p)(x+c^p)$ associated with the FLT solution $a^p + b^p = c^p$: \begin{itemize} \item The discriminant, which is equal to the product of the squares of all the differences of the roots. For the Frey curve, it is equal to $$( - b^p - 0 )^2 ( - c^p - 0 )^2 ( c^p - b^p)^2$$ which by our assumption that $a,b,c$ is a solution to FLT, is equal to $a^{2p}b^{2p}c^{2p}$. \item The conductor, $N = \prod_{p \mid abc } p$. For a modular elliptic curve, this $N$ is the same as the level $N$ of the curve's modular form. \item The $j$-invariant, $j = 2^8 \frac{(b^{2p} + c^{2p} - b^p c^p )}{a^{2p}b^{2p}c^{2p}}$ \end{itemize} We are now in a position to define what semistable means. When a prime $l$ divides the discriminant of an elliptic curve, at least two of the roots become congruent mod $l$. An elliptic curve is semistable if for all such primes $l$, only two roots becomes congruent mod $l$. The Frey curve is semistable since the discriminant is $a^{2p}b^{2p}c^{2p}$, the roots are 0, $-b^p$, and $-c^p$, and $b^p$ and $c^p$ are relatively prime. The following string of results prove FLT: \begin{theorem}[Wiles] Semistable elliptic curves over the rational numbers are modular. \end{theorem} \begin{corollary} Frey curves are modular. \end{corollary} \begin{lemma} The $j$-invariant of Frey curves is exactly divisible by $p^{m p}$, where $m p$ is some multiple of $p$. \end{lemma} \bpf\ This is an easy exercise which is left to the reader. \medskip In the context of these three results --- modular elliptic curves whose $j$-invariants are exactly divisible by $p^{mp}$ --- the Serre conjecture (proved by Ribet) states that if we have a modular form $F$ of level $N$, then for ALL odd primes $l$ dividing $N$, there is a modular form $F'$ of level $N/l$. [The Serre conjecture also requires a particular kind of Galois representation on the Frey curve, but we won't discuss this other than to say that the Frey curve has it.] \begin{theorem} The equation $x^p + y^p = z^p$ has no solutions with $a,b,c$ non-zero for $p$ an odd prime. \end{theorem} \bpf\ Say there was a solution $a^n + y^n = z^n$. Then we have a Frey curve, and this Frey curve has a modular form $F$ of weight 2 and level $N$ (a cusp form). By the Serre conjecture, for any odd prime $l$ dividing N, there is a modular form $F'$ of level $N/l$ with $$ F' \equiv F \pmod{p}$$ But then, if $l'$ is another odd prime dividing $N$, we can apply Serre's conjecture again to $F'$ and get a modular form $F''$ with even smaller level $N/l l'$, and then apply it again to $F''$, etc. Eventually we get a modular form $\tilde{F}$ of weight 2 and level 2, and $\tilde{F}$ is again a cusp form. [Note that $2$ must divide one of $a$, $b$, and $c$, and hence $2 \mid N$.] But there are no cusp forms of weight 2 and level 2, a contradiction. \nl \begin{center} {\bf \LARGE Q.E.D.} \end{center} \end{document} -- David Radcliffe radcliff@csd4.csd.uwm.edu &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& From: radcliff@csd4.csd.uwm.edu (David G Radcliffe) Newsgroups: sci.math Subject: Re: Fermat's Last Theorem Talk Date: 21 Jul 1993 03:19:27 GMT A few days ago, I posted on sci.math a lecture by David Cox. I did this without the author's permission or knowledge, and I would like to apologize publicly for my lack of consideration. Also, I would like to thank Prof. Cox for his lucid explanation of the ideas underlying Wiles' proof. David Cox mailed the following addendum to me. He asked me to forward it to sci.math. -- David Radcliffe radcliff@csd4.csd.uwm.edu ---------------------------CUT HERE----------------------------- \documentstyle[12pt]{article} \setlength{\textheight}{9in} \addtolength{\textwidth}{1in} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \addtolength{\topmargin}{-.85in} \title{{\sc Addendum to} Fermat's Last Theorem} \author{David Cox (Amherst College)} \begin{document} \date{July 20, 1993} \maketitle A transcript of a lecture I gave on Fermat's Last Theorem was recently posted on {\tt sci.math}. The transcript was not intended for wide dissemination and contained errors and omissions, both historical and mathematical. This addendum addresses the problems that were present in the original. I am grateful to my colleagues who pointed out the errors. \begin{itemize} \item The second part of the lecture (on the period from Euler to Faltings) explicitly stated that I was only giving highlights. The third part (on the period from Faltings to Wiles) should have had a similar disclaimer. My discussion of recent developments only touched on the high points of what's been happening in this area. \item One omission in the lecture is the fact that dozens of people have contributed to the theory of elliptic curves, galois representations and modular forms. It would be nice if someone would prepare a bibliography of relevant books and papers. \item 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. \item As for the events in 1988 concerning the arithmetic BMY inequality, it would be more accurate to say that Miyaoka gave a lecture in Bonn in which he stated the inequality as a theorem, which would imply FLT for large $p$. 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. \item These days, Taniyama--Weil conjecture is usually called the Taniyama--Shimura conjecture. See pages 131--132 of Lang's {\sl Number Theory III: Diophantine Geometry} (Springer, 1991) for a discussion of this conjecture. \item In the discussion of the modular form $F(z)$, it should be pointed out that $F(z)$ is an eigen-form for a certain Hecke algebra acting on all cusp forms of weight 2 and level $N$. \item It is not quite correct to say that Frey proved that FLT follows from Taniyama--Shimura and a conjecture of Serre. The actual story is more complicated. In 1985, Frey tried to prove that Taniyama--Shimura implies FLT, but there were serious gaps in his proof. Several people tried to fix Frey's argument, but it was Serre who saw that the a conjecture on level reduction would fill the gap. Hence it is better to say that Frey and Serre proved that FLT follows from Taniyama--Shimura and a conjecture of Serre. \item There is also a 1987 conjecture of Vojta concerning heights of points (relative to the canonical class) on a curve defined over a number field. This conjecture implies the Mordell conjecture, and it also implies FLT for large exponents. See pages 63--64 of Lang's book (cited above) for more details. Hence, by 1990, there were {\em three} ways to prove FLT using standard conjectures in number theory. \item When working with a solution $a^p + b^p = c^p$ of FLT, we always assume that $a,b,c$ are nonzero and that $p$ is a prime $> 3$ (the condition $p > 3$ is needed in order to satisfy the hypothesis of Serre's conjecture). \item Note that a solution $a^p + b^p = c^p$ gives solutions $b^p + a^p = c^p$ and, since $p$ is odd, $a^p + (-c)^p = (-b)^p$. From here, it is easy to rearrange the solution so that (after relabeling) we have $b$ even and $c \equiv 1 \bmod 4$. This is needed in order for the Frey curve to be semistable at 2. \item In working with elliptic curves over {\bf Q}, one needs to distinguish between the discriminant and the {\em minimal discriminant}. For the Frey curve, the minimal discriminant is $2^{-8}a^{2p}b^{2p}c^{2p}$. Since $b$ is even and $p \ge 5$, this is still an integer. \item The definition of semistable given in the lecture only works for primes bigger than $3$, and similarly the proof only works for these primes. More care is needed in showing that the Frey curve is semistable at 2 and 3. In particular, the conditions $b$ even and $c \equiv 1 \bmod 4$ are needed for the curve to be semistable at 2. \item The statement of Lemma 1 is not strong enough. The correct statement should that that if $l$ is an odd prime dividing the conductor $N$, then the $j$-invariant can be written as $l^{mp}$ times a fraction not involving $l$. \item The paragraph following the proof of Lemma 1 should state: ``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 conjecture of Serre (to be discussed below) now applies for {\em all} odd primes dividing the conductor. We can now prove Fermat's Last Theorem:'' \item Replace the proof of Theorem 2 with the following: ``Say there was a solution to $a^p+b^p = c^p$. Then we have a Frey curve, which as we know has a modular form $F$ of weight 2 and level $N$ (a cusp form). This curve also has a Galois representation $\rho$ on 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 conjecture are satisfied for all odd primes $l$ dividing $N$. In such a case, the conjecture (proved by Ribet) asserts that there is a modular form $F'$ of level $N/l$ with $$F' \equiv F \bmod p$$ This 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 Serre's conjecture again to $F'$ and get a modular form $F''$ with even smaller level $N/l l'$, and then apply it again to $F''$, etc. Eventually we get a modular form $\tilde{F}$ of weight 2 and level 2, and $\tilde{F}$ is again a cusp form. [Note that $2$ must divide one of $a$, $b$ or $c$, and hence $2\mid N$]. But there are no cusp forms of weight 2 and level 2, a contradiction.'' \end{itemize} I should also explain how my lecture came to be posted on {\tt sci.math}. I was invited to give a lecture on FLT to the Regional Geometry Institute being held at Smith College in July, 1993. The lecture was to be an ``All-Institute'' event, which meant that the audience consisted of high school teachers, undergraduates, graduate students, and researchers in discrete and computational geometry. Hence the lecture was informal with few mathematical prerequisites. As with the other ``All-Institute'' events, a transcript was posted on an electronic geometry forum. Due to a mix-up, the version posted was not the final one, which accounts for some (but not all) of the errors. From here, the lecture took on a life of its own and moved to {\tt sci.math} without my knowledge. But this is how things work in the electronic age, and I take full responsibility for what happened. It is important to note that even with the above corrections and additions, what I've written is far from the complete story of the proof of FLT. In fact, this addendum may be regarded as a invitation for someone to write a fuller account of what happened. I can assure you that there will be an interested audience. An expanded and corrected version of my original lecture (which incorporates all of the above changes) has been posted on the AMS gopher. For most Unix systems, you can access easily this by entering {\tt gopher e-math.ams.org} \noindent at your system prompt. If you don't have gopher on your system, you can try {\tt telnet e-math.ams.org} \noindent and give {\tt e-math} as both login and password. Then the e-math menu will show you how to start up the gopher. Once in the AMS gopher, go to item 7, titled ``Mathematical Discussions Lists and Bulletin Boards''. The revised version of the lecture is listed as ``Introduction to 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}