NUMBER THEORY AS GADFLY - BARRY MAZUR ROBERT C. RHOADES Abstract. This paper appeared in the American Mathematical Monthly in August- September 1991 pages 593-610. This paper gives a very general overview of how what the Shimura-Taniyama conjecture is and how the truth of it can be used to prove Fermat s last theorem. For me this is very useful because I have always used Fermat s Last Theorem as the motivating result when describing the modularity of elliptic curves to nonmathematicians. (Which I do more often then anyone might think.) For my own use I will try to summarize the statement of the Shimura-Taniyama(-Weil) conjecture and also how it implies Fermat s Last Theorem. I point out that Mazur says little more about how it applies Fermat s Last theorem then I do, however his description of the S-T-W conjecture is the bulk of the middle ten pages of his article. As a side note, Mazur s article was written before Wiles announced his proof. 1. Comments on Fermat s Last Theorem Number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort! -Mazur One of those bugs is Fermat s last theorem. It certainly bit Andrew Wiles who spent most of his life trying to find a proof of FLT. I think what is interesting about Fermat s Last theorem is that it has little application to other problems in mathematics. This is very much unlike other problems that motivated number theory for a long time. For example, the Riemann Hypothesis has motivated large branches of number theory for a long time, but there is something essentially different about the two problems. I point out two interesting differences. For both problems it is not the truth of problem that is all that important. It is not the truth of the Riemann Hypothesis that is all that important but instead the implications of the truth of the Generalized Riemann Hypothesis that are truly important. That is to say that the Riemann hypothesis was just a first step toward the realization of a much more general theorem, the more general theorem has tremendous important to other problems. Sarnak made a comment about this in his first lecture at the Fields Institute in 2002. On the other hand, for Fermat there is no more general question that naturally fits this theorem, of which the more general problem has significance to other branches of math and science. I point out that you can Date: December 10, 2005. 1
2 ROBERT C. RHOADES consider Fermat s Last Theorem of a specific example of the more general Diophantine equation AX n + BY n = CZ n, which Mazur does briefly consider. As a second falling out, for the Riemann hypothesis there are many results that assume its truth, but you never see theorems starting with Assuming Fermat s Last Theorem. Mazur writes, Despite the fact that its truth hasn t a single direct application (even within number theory!) it has, nevertheless, an interesting oblique contribution to make to number theory: its truth would follow from some of the most vital and central conjectures in the field. So we see that, unlike the case of the Riemann Hypothesis FLT is not important because of the result, but instead because of the techniques that went into proving it... the methods that were created as a result of its study are the true value of the theorem. I do not intend to say that the methods developed to study the Riemann hypothesis are not important, but I simply point out that somehow there is a fundamental difference between these two problems. The main conjecture that implies the truth of FLT is the Shimura-Taniyama-Weil Conjecture which is sometimes referred to as just the Shimura-Taniyama conjecture. As we shall see, the conjecture of Shimura-Taniyama-Weil would imply a strange and important connection between elliptic curves that arise in Arithmetic and the Hyperbolic Plane. One of the things that Mazur, unfortunately does not really explain in this article is how the hyperbolic plane is related to modular forms, and hence how the Shimura- Taniyama conjecture is someone the same as the modularity of elliptic curves, which is a phrase that gets tossed around a lot. Before moving on we quote Mazur as to the confusion about what to call the conjecture: The fact that this conjecture has also been referred to as the Wiel Conjecture, the Taniyama-Weil conjecture, and the Taniyama conjecture points to the difficulty in assigning it a clear attribution. It was originally formulated as a problem of Taniyama in a conference in 1955 and was published in Japanese. A more precise formulation corresponding to the modern form of the conjecture-involving important information concerning the conductor - was implicitly suggested by subsequent work of Weil which had the effect of bringing the problem to the attention of a large audience. The most precise version of this conjecture to date, which brings the crucial issue of fields of definition, incorporates work of Shimura, Eichler, and others. 2. Another Result to Look-up This paper was the first that I read about a nice result of Faltings. According to Mazur, the theorem of Faltings (conjectured originally by Mordell) which asserts that any algebraic equation in two variables, and of genus greater than or equal to 2, has only a finite number of rational solutions. If an elliptic curve has finitely many points, then its group of rational points, which is finitely generated and abelian, has rank 0. It is conjectured, but not proved that
NUMBER THEORY AS GADFLY - BARRY MAZUR 3 half of all elliptic curves have rank 0 and half have rank 1. Of course, each elliptic curves is associated to a torus and hence has genus 1. So this theorem does not apply to them. 3. Euclidean Uniformization The basic idea of a Euclidean uniformization is that it gives us a map from the complex numbers with the Euclidean metric to the set of elliptic curves. The idea here is that we begin with a lattice of C, we call the lattice Λ. Then C/Λ is a torus. The map from the complex plane to C/Λ is called the covering map. In this case we can think of the torus as inheriting a conformal geometry from the Euclidean geometry of C. With this geometry we think of the torus as a Riemann surface. We then ask for the functions on this Riemann surface, which by construction are doubly periodic functions on C. It turns out that these doubly periodic functions, which I think of as being of the form (z) := 1 z 2 + ω 0 ( 1 (z ω) 2 1 ω 2 with the ω Λ, are precisely the meromorphic functions on C/Λ. It is a beautiful result that these functions satisfy a differential equation of the form ( ) 2 = 4 3 + A + B, where A and B are complex numbers. The point now is that Y 2 = 4X 3 + AX + B is an algebraic curve in the (X, Y ) plane, this is an elliptic curve. [[Any degree three curve can be changed into this form, which is called the Weierstrass form.]] So we see that the problem of finding the solutions in complex numbers to this solution is the same as finding the function and using (X, Y ) = ( (z), (z), where z ranges over C \ Λ. Completing the picture we obtain a map from C points on the elliptic curve. Remark. It is important that this construction, because is a function on the torus, allows us to identify the complex points of the elliptic curve with the torus! The map is the covering mapping. Definition 3.1. So we say that an elliptic curve has a Euclidean uniformization when we have such a mapping from the complex numbers to the elliptic curve as described above. More formally we have the following definition. Definition 3.2. A hyperbolic uniformization of E is defined to be a covering mapping periodic with respect to some subgroup of finite index in Γ from C to the complex points of the elliptic curve. ),
4 ROBERT C. RHOADES 4. Hyperbolic Uniformization In the previous section we constructed a theory which allowed us to associate in a precise way a quotient of C with the Euclidean geometry with the complex points of an elliptic curve. We might hope to do the same but with a different geometry on the complex numbers. In particular we begin with the model of hyperbolic geometry which consists of the upper-half plane where the straight lines are the circles perpendicular to the real line and the straight lines with constant real part. Notice that in hyperbolic geometry parallel lines may intersect. In Euclidean geometry we formed a lattice by having two translations which were independent. Here we think of our two symmetries, not just as translations, but we may also have a sort of reflection. That is instead of considering z z + λ 1 and z z + λ 2 we have the maps T : z z + 1 and S : z 1/z. Think of T and S as 2 2 matrices we know that S and T generate Γ = SL 2 (Z). Analogously to our construction from before, Definition 4.1. Our hyperbolic uniformization would be a mapping from H E, where H is the upper-half plane and E is the complex points of the elliptic curve, which is a covering map and which is periodic with respect to a subgroup of finite index in Γ. It is a theorem of Bely that says that an elliptic curve admits a hyperbolic uniformization if and only if it has a Weierstrass equation with coefficients A, B which are algebraic numbers, i.e. A, B Q. The content of the Shimura-Taniyama-Weil conjecture is that if we restrict further the coefficients of our elliptic curve, then we can say more about the subgroup of Γ for which the covering map is periodic. 5. Shimura-Taniyama-Weil We give the following definitions which will be important in our discussion. Definition 5.1. An arithmetic elliptic curve is one whose defining equation can be taken with coefficients in Q. This will be the restriction we place on the elliptic curve. The next definition describes what the subgroup of Γ should be for the hyperbolic uniformization of these elliptic curves. Definition 5.2. Let E be an elliptic curve. A hyperbolic uniformization of E of arithmetic type is a hyperbolic uniformization of the elliptic curve which is periodic with respect to a congruence subgroup Γ Γ.
NUMBER THEORY AS GADFLY - BARRY MAZUR 5 Although any elliptic curve admits a Euclidean uniformization and an elliptic curve admits a hyperbolic uniformization if and only if it can be defined by a Weierstrass equation with coefficients A, B which are algebraic numbers, the Shimura- Taniyama - Weil conjecture asserts the following: Conjecture 5.3 (Shimura-Taniyama-Weil Conjecture). Any arithmetic elliptic curve admits a hyperbolic uniformization of arithmetic type. Of course this conjecture is now a theorem as a result of the work of Wiles, Conrad, Taylor and others. Mazur draws a nice picture of a Euclidean plane and a hyperbolic plane and arrows from each pointing to a torus. As Mazur writes, It is the confluence of two uniformizations, the Euclidean one, and the (conjectural) hyperbolic one of arithmetic type, that puts an exceedingly rich geometric structure on an arithmetic elliptic curve, and that carries deep implications for arithmetic questions. 6. Shimura-Taniyama-Weil implies Fermat Suppose for a contradiction that we have a prime p > 150, 000 and a, b, c Z with a p + b p = c p. Then we can define an elliptic curve y 2 = x(x a p )(x + b p ). It was suggested by Frey, who considered the arithmetic of the curve, that this curve is does not admit a hyperbolic uniformization of arithmetic type or equivalently that this curve is not modular. Ribet established the result. Since the Shimura-Taniyama conjecture implies that every elliptic curve is modular we would obtain a contradiction and therefore establish that there are no such a, b, and c. I do not understand the proof of this result, but Mazur gives a quick overview and references the Bourbaki report of Osterlé, Nouvelles approaches du Theoreme de Fermat. Department of Mathematics, University of Wisconsin, Madison, WI 53706 E-mail address: rhoades@math.wisc.edu