ADHONOREMSIRANDREWJ.WILES

Wiles received the Abel Prize from Crown Prince Haakon of Norway.

Wiles ultimately established in a proof both surprising and profound. It is especially appropriate that Wiles’s groundbreaking work on elliptic curves was recognized by the awarding of the Abel Prize, elliptic curves being the natural domains of the elliptic functions introduced n May 24, 2016, Sir Andrew J. Wiles received by Abel. As the citation for the 2016 Abel Prize concludes: the Abel Prize in a ceremony held in the Aula of the University of Oslo in Oslo, Norway. Wiles, “Few results have as rich a mathematical history and as who received the prize from H.R.H. Crown dramatic a proof as Fermat’s Last Theorem.” Prince Haakon at the award ceremony, was The awarding of the Abel Prize was followed by the Othe fourteenth recipient of the 6 million NOK (about Abel Lectures on the next day, May 25. In his lecture, 750,000 USD) prize. A prize honoring the Norwegian Wiles explained how his proof of Fermat’s Last Theorem mathematician Niels Henrik Abel was first proposed by exemplified the movement of number theory from the the world-renowned mathematician Sophus Lie, also from abelian to the nonabelian. Henri Darmon, in his lecture Norway, and initially planned for the one-hundredth an- “Andrew Wiles’s Marvelous Proof,” described Wiles’s work niversary of Abel’s birth in 1902, but the establishment as “a centerpiece of the Langlands program” and ex- of the Abel Prize had to wait another hundred years. The plained its transformative impact, and Manjul Bhargava Abel Prize is administered by the Norwegian Academy of spoke about how Wiles’s work has implications for the Science and Letters. Birch–Swinnerton-Dyer Conjecture. The Abel Prize was awarded to Wiles for “his stunning proof of Fermat’s Last Theorem,” which opened a new era Photo Credits in number theory. The citation of the Abel Committee, Photos are courtesy of Audun Braastad. read by committee chair John Rognes on the occasion of the announcement of the 2016 Abel Prize, recounts the early history of Fermat’s Last Theorem—the assertion that The full text of the citation of the Abel Committee for any given integer 푛 ≥ 3, there are no integer solutions can be found at www.abelprize.no/c54154/binfil/ to 푥푛 + 푦푛 = 푧푛 with 푥푦푧 ≠ 0—and how it was eventually download.php?tid=67039. An expanded version of linked to the then-conjectural modularity of semistable Henri Darmon’s Abel Lecture is included in this issue. elliptic curves and that it was this modularity that Andrew DOI: http://dx.doi.org/10.1090/noti1486

March 2017 Notices of the AMS 197 Interview with Abel Laureate Sir Andrew J. Wiles Martin Raussen and Christian Skau

Raussen and Skau : Professor Wiles, please accept our con- gratulations for having been selected as the Abel Prize Laureate for 2016. To be honest, the two of us expected this interview to take place several years ago! You are famed not only among mathematicians but also among the public at large for (and now we cite the Abel Committee): “the stunning proof of Fermat’s Last Theorem, by way of the Modularity Conjecture for elliptic curves, opening a new era in number theory.” This proof goes back to 1994, which means that you had to wait for more than twenty years before it earned you the Abel Prize. Neverthe- less, you are the youngest Abel Prize Laureate so far. After you finished your proof of Fermat’s Last Theorem you had to undergo a deluge of interviews, which makes our task difficult. How on earth are we to come up with questions that you have not answered several times before? Well, we will try our best. Sir Andrew J. Wiles received the Abel Prize from Crown Prince Haakon of Norway. Fermat’s Last Theorem: A Historical Account We have to start at the very beginning, with a cita- this margin is too narrow to contain.” This remark was tion in Latin: “…nullam in infinitum ultra quadratum written in the year 1637 by the French lawyer and amateur potestatem in duos eiusdem nominis fas est dividere,” mathematician Pierre de Fermat [1601–1665] in the mar- which means: “it is impossible to separate any power gin of his copy of Diophantus’ Arithmetica. He certainly higher than the second into two like powers.” In mod- did not expect that it would keep mathematicians, profes- ern mathematical jargon, this can be written: “The sionals, and amateurs alike busy for centuries trying to equation 푥푛 + 푦푛 = 푧푛 has no solution in natural numbers unearth the proof. for 푛 greater that two.” And then it continues: “cuius rei Could you please give us a short account of some of demonstrationem mirabilem sane detexi. Hanc marginis the attempts towards proving Fermat’s Last Theorem up exiguitas non caperet,” which means: “I have discovered a until the time you embarked on your successful journey? truly marvellous proof of this, which Furthermore, why was such a simple-minded question so attractive, and why were attempts to prove it so productive in the development of number theory? Wiles: The first serious attempt to solve it was pre- Martin Raussen is professor of mathematics at Aalborg Univer- sumably by Fermat himself. But, unfortunately, we know sity, Denmark. His e-mail address is [email protected]. nothing about it except for what he explained about his Christian Skau is professor of mathematics at the Norwegian proofs in the specific cases of 푛 = 3 and 푛 = 4.1 That is, University of Science and Technology, Trondheim, Norway. His he showed that you can’t have the sum of two cubes be e-mail address is [email protected]. another cube or the sum of two fourth powers being a For permission to reprint this article, please contact: [email protected]. 1Strictly speaking, Euler was the first to spell out a complete proof DOI: http://dx.doi.org/10.1090/noti1484 in the case 푛 = 3.

198 Notices of the AMS Volume 64, Number 3 fourth power. He did this by a beautiful method, which contradict another well-known conjecture called the Mod- we call infinite descent. It was a new method of proof, or ularity Conjecture. This conjecture had been proposed in at least a new way of presenting proofs in arithmetic. He a weak form by Taniyama [1927–1958] and was refined explained this method to his colleagues in letters, and he by Shimura, but the first real evidence for it came from also wrote about it in his famous margin, which was big André Weil [1906–1998], who made it possible to check enough for some of it at least. After the marginal notes this precise form of the Modularity Conjecture in some were published by Fermat’s son after his father’s death, detail. And a lot of evidence was amassed showing that it lay dormant for a while. Then it was picked up by Euler this should certainly be true. So, at that point, mathe- [1707–1783] and others who tried to find this truly mar- maticians could see that: “Yes, Fermat is going to be true. vellous proof. And they failed. It became quite dramatic Moreover, there has to be a proof of it.” What happened in the mid-nineteenth century—various people thought was that the Modularity Conjecture was a problem that they could solve it. There was a discussion concerning mathematics could not just put to one side and go on for this in the French Academy: Lamé [1795–1870] claiming he was just about to prove it, and Cauchy [1789–1857] another five hundred years. It was a roadblock right in the saying he thought he could too, and so on. middle of modern mathematics. It was a very, very central In fact, it transpired that the German mathematician problem. As for Fermat, you could just leave it aside and Kummer [1810–1893] had already written a paper where forget it almost forever. This Modularity Conjecture you he explained that the fundamental problem was what is could not forget. So, at the point when I heard that Ribet known now as the fundamental theorem of arithmetic. In had done this, I knew that this problem could be solved our normal number system, any number can be factorized and I was going to try. in essentially one way into prime factors. Take a number Raussen and Skau : Concerning speculations about Fer- like 12; it is 2 times 2 times 3. There is no other way of mat’s claimed proof, do you think he had the same idea as breaking it up. But in trying to solve the Fermat problem, Lamé, assuming, wrongly as it turned out, that the cyclo- you actually want to use systems of numbers where this tomic integers have unique factorization? uniqueness does not hold. Every attempt that was made Wiles: No, I don’t think so, though the idea might be to solve the Fermat problem had stalled because of this in there somewhere. It is very hard to understand. André failure of unique factorization. Kummer analyzed this in Weil wrote about this. All the other problems Fermat incredible detail. He came up with the most beautiful considered had to do with curves that were of genus results, and the end product was that he could solve zero or genus one. And suddenly he is writing down a it for many, many cases. For example, for 푛 ≤ 100 he curve that has higher genus. How is he going to think solved it for all primes except for 37, 59, and 67. But about it? When I was trying this myself as a teenager, he did not finally solve it. His method was based on the I put myself in Fermat’s frame of mind because there idea that Fermat had introduced—the method of infinite was hardly anything else I could do. I was capable of descent—but in these new number systems. The new number systems he was using spawned understanding his mathematics from the seventeenth algebraic number theory as we see it today. One tries to century but probably not much beyond that. It seemed to solve equations in these new systems of numbers instead me that everything he did came down to something about of solving them with ordinary integers and rational quadratic forms, and I thought that might be a way of numbers. Attempts in the style of Fermat carried on for a trying to think about it. Of course, I never succeeded, but while but somewhat petered out in the twentieth century. there is nothing else that suggests Fermat fell into this No one came up with a fundamentally new idea. In the trap with unique factorization. In fact, from the point of second half of the twentieth century, number theory view of quadratic forms, he understood that sometimes moved on and considered other questions. Fermat’s there was unique factorization and sometimes there was problem was all but forgotten by the professionals. not. So he understood that difference in his own context. Then, in 1985, Ger- I think it is unlikely that that was the mistake. hard Frey, a German Raussen and Skau : In the same book by André Weil mathematician, came up It was a that you referred to, titled Number Theory: An approach with a stunning new through History from Hammurapi to Legendre, it is men- idea where he took a roadblock right in tioned that Fermat looked at the equation of a cube minus hypothetical solution to the middle of a square equal to 2 [푥3 −푦2 = 2] and he showed that it has the Fermat problem and essentially only one solution, namely 푥 = 3 and 푦 = ±5. rewrote it so that it made André Weil speculates that Fermat at the time looked at what is called an elliptic modern the ring 푍[ − 2], which does have unique factorization. curve. And he showed, mathematics. √ or suggested, that this Wiles: Yes, he used unique factorization, but the way he elliptic curve had very did it was in terms of quadratic forms. And I think he also peculiar properties. He conjectured that you can’t really looked at quadratic forms corresponding to 푍[√ − 6] have such an elliptic curve. Building on this a year later, an where there is not unique factorization. So I think he American mathematician, Kenneth Ribet, demonstrated, understood. It was my impression when I thought about using this Frey curve, that any solution of Fermat would it that he understood the difference.

March 2017 Notices of the AMS 199 A Mathematical Education allowed to do number theory in your first year. And you Raussen and Skau : You were apparently already inter- could not really get down to it before your third year. ested in mathematical puzzles as quite a young boy. Have Raussen and Skau : But you were not interested in ge- you any thoughts about where this interest came from? ometry, not as much as in algebra and number theory, Were you influenced by anyone in particular? anyway? Wiles: I just enjoyed mathematics when I was very Wiles: No, I was primarily interested in algebra and young. At the age of ten, I was looking through library number theory. I was happy to learn these other things, shelves devoted to mathematics. I would pull out books but I really was most excited about number theory. My and at one point I pulled out a book of E. T. Bell teachers arranged for me to take extra classes in number [1883–1960] titled The Last Problem, which on its cover theory, but there was not that much to offer. describes the Fermat equation, the Wolfskehl Prize, and At one point, I decided that I should put all the years the romantic history of the problem. I was completely of Latin I had done at school to good use and try to captivated by it. read some of Fermat in the original, but I found that was Raussen and Skau : Were there other things that fasci- actually too hard. Even if you translated the Latin, the way nated you in this book by Eric Temple Bell? they wrote in those days wasn’t in the algebraic symbols Wiles: It is entirely about that one equation, really. And I was used to, so it was quite difficult. it is actually quite wordy. So there is less mathematics in Raussen and Skau : It must have been a relief when you were done and came to Cambridge to start studying some sense than you might think. I think it was more the number theory for real, with John Coates as your supervi- equation. Then, when I found this equation, I looked for sor. other elementary books on number theory and learned Wiles: That’s right. I had a year, a preliminary year, about congruences and solved congruences and so on, in which I just studied a range of subjects and then I and looked at other things that Fermat did. could do a special paper. John Coates was not yet at Raussen and Skau : You did this work besides your Cambridge, but I think he helped me, maybe over the ordinary schoolwork? summer. Anyway, that summer I met him and started Wiles: Yes, I don’t think my schoolwork was too taxing working with him right away, and that was just wonderful. from that point of view. The transition from undergraduate work, where you were Raussen and Skau : Was it already clear to you at that just reading and studying, to research—that was the real time that you had an extraordinary mathematical talent? break for me. It was just wonderful. Wiles: I certainly had a mathematical aptitude and obviously loved to do mathematics, but I don’t think I felt Elliptic Curves that I was unique. In fact, I don’t believe I was unique in Raussen and Skau : We assume it was John Coates who the school I attended. There were others who had just as initiated your work on elliptic curves and Iwasawa theory? strong a claim to be future mathematicians, and some of Wiles: Absolutely. He had some wonderful ideas and them have become mathematicians, too. was generous to share them with me. Raussen and Skau : Did you already plan to study math- Raussen and Skau : Did you tell John Coates that you ematics and to embark on a mathematical career at that were interested in the Fermat problem? age? Wiles: Perhaps I did. I don’t remember. It is really true Wiles: No, I don’t think I really understood you could that there hadn’t been any new ideas since the nineteenth spend your life doing mathematics. I think that only came century. People were trying to refine the old methods and, later. But I certainly wanted to study it as long as I could. yes, there were refinements. But it didn’t look like these I’m sure that as far as my horizon extended, it involved refinements and the solution were going to converge. It mathematics. was just too hard that way. Raussen and Skau : You started to study mathemat- Raussen and Skau : At the time you started to work ics as a student at Oxford in 1971. Can you tell us a little with John Coates, you had no idea that these elliptic curves bit about how that worked out? Were there any particu- were going to be crucial for the solution of Fermat’s Last lar teachers or any particular areas that were particularly Theorem? important for you? Wiles: No, it’s a wonderful coincidence. The strange Wiles: Before I went to college (actually in high school), thing is that, in a way, the two things that are most one of my teachers had a PhD in number theory. He prominent in Fermat that we remember today are his gave me a copy of Hardy and Wright’s An Introduction work on elliptic curves and his famous last theorem. For to the Theory of Numbers, and I also found a copy of example, this equation you mentioned, 푦2 + 2 = 푥3, is an Davenport’s The Higher Arithmetic. And these two books elliptic curve. And the two strands came together in the I found very, very inspiring in terms of number theory. proof. Raussen and Skau : So you were on track before you Raussen and Skau : Could you explain what an elliptic started studying? curve is and why elliptic curves are of interest in number Wiles: Yes, I was on track before. In fact, to some theory? extent, I felt college was a distraction because I had to Wiles: For a number theorist, the life of elliptic curves do all these other things: applied maths, logic, and so on, started with Fermat as equations of the form 푦2 equals and I just wanted to do number theory. You were not a cubic polynomial in 푥 with rational coefficients. Then,

200 Notices of the AMS Volume 64, Number 3 the problem is to find the rational solutions to such an equation. What Fermat noticed was the following. Sometimes you can start with one or even two rational solutions and use them to generate infinitely many others. And yet sometimes there are no solutions. This latter situation occurs, for example, in the case 푛 = 3 of Fermat’s Last Theorem, the equation being, in fact, an elliptic curve in disguise. Sometimes you can show there are no rational solutions. You could have infinitely many and you could have none. This was already apparent to Fermat. In the early nineteenth century, one studied these equations in complex numbers. Abel [1802–1829] himself came in at this point and studied elliptic functions and related these to elliptic curves, implying that elliptic curves have a group structure. They were very well Interviewed in Oslo in May, Wiles told Martin understood in terms of doubly periodic functions in the Raussen and Christian Skau that he set out to prove early nineteenth century. But that is what underlies the the Modularity Conjecture with no idea from what complex solutions, solutions to the equation in complex branch of mathematics the answer would come. numbers. The solutions to the equation in rational numbers were studied by Poincaré [1854–1912]. What’s now known as Raussen and Skau : Was this the first general result the Mordell–Weil theorem was proved by Mordell [1888– concerning the Birch and Swinnerton-Dyer Conjecture? 1972] and then Weil in the 1920s, answering a question of Wiles: It was the first one that treated a family of cases Poincaré. In our setting, it says that the 퐾-rational points rather than individual cases. There was a lot of numerical on an elliptic curve over a number field 퐾, in particular for data for individual cases, but this was the first infinite 퐾 equal to the rationals, form a finitely generated abelian family of cases. group. That is, from Fermat’s language, you can start with Raussen and Skau : This was over the rational num- a finite number of solutions and, using those, generate bers? all the solutions by what he called the chord-and-tangent Wiles: Yes. process. Raussen and Skau : We should mention that the Birch By now you know the structure; it is a very beautiful and Swinnerton-Dyer Conjecture is one of the Clay Mil- algebraic structure, the structure of a group, but that lennium Prize Problems, which would earn a person who does not actually help you find the solutions. So, no one solves it one million dollars. really had any general methods for finding the solutions Wiles: That’s right. I think it’s appealing, partly be- until the conjectures of the 1960s, which emerged from cause it has its roots in Fermat’s work, just like the the Birch and Swinnerton-Dyer Conjecture. There are two Fermat problem. It is another “elementary-to-state” prob- aspects to it; one is somewhat analytic, and one is in terms lem concerned with equations—in this case of very low of what is called the Tate–Shafarevich group. Basically, degree—which we can’t master and which Fermat initiated. the Tate–Shafarevich group gives you the obstruction to I think it is a very appealing problem. an algorithm for finding the solutions. And the Birch Raussen and Skau : Do you think it is within reach? In and Swinnerton-Dyer Conjecture tells you that there is other words, do we have the necessary tools for somebody actually an analytic method for analyzing this so-called daring enough to attack it and succeed? Or do we have to Tate–Shafarevich group. If you combine all this together, wait for another three hundred years to see it solved? ultimately it should give you an algorithm for finding the Wiles: I don’t suppose it will take three hundred years, solutions. but I don’t think it is the easiest of the Millennium Problems. I think we are still lacking something. Whether Birch and Swinnerton-Dyer, Tate–Shafarevich, the tools are all here now, I am not sure. They may be. Selmer… There are always these speculations with these really Raussen and Skau : You worked on the Birch and difficult problems; it may be that the tools simply aren’t Swinnerton-Dyer Conjecture when you were a graduate there. I don’t believe that anyone in the nineteenth century student together with John Coates? could have solved Fermat’s Last Theorem, certainly not in Wiles: Yes, that is exactly what he proposed working the way it was eventually solved. There was just too big on. We got the first result in certain special families of a gap in mathematical history. You had to wait another elliptic curves on this analytic link between the solutions hundred years for the right pieces to be in place. You can and what is called the 퐿-function of the elliptic curve. never be quite sure about these problems, whether they Raussen and Skau : These were curves admitting com- are accessible to your time. That is really what makes plex multiplication? them so challenging; if you had the intuition for what can Wiles: Exactly; these were the elliptic curves with be done now and what can’t be done now, you would be complex multiplication. a long way towards a solution!

March 2017 Notices of the AMS 201 Raussen and Skau : You mentioned the Tate–Shafarevich The Modularity Conjecture group and in that connection the Selmer group appears. Raussen and Skau : What you proved in the end was a Selmer [1920–2006] was a Norwegian mathematician, special case of what is now called the Modularity Conjec- and it was Cassels [1922–2015] who was responsible for ture. In order to explain this, one has to start with modular naming this group the Selmer group. Could you say a few forms and how modular forms can be put in relation with words about the Selmer group and how it is related to the elliptic curves. Could you give us some explanations? Tate–Shafarevich group, even if it’s a little technical? Wiles: Yes; we have described an elliptic curve (over 2 3 Wiles: It is technical, but I can probably explain the the rationals) as an equation 푦 = 푥 +푎푥+푏, where the 푎 basic idea of what the Selmer group is. What you are and 푏 are assumed to be rational numbers. (There is also a condition that the discriminant should not vanish.) As I trying to do is to find the rational solutions on an said, at the beginning of the nineteenth century you could elliptic curve. The method is to take the rational points describe the complex solutions to this equation. You could on the elliptic curve—suppose you have got some—and describe these very nicely in terms of the Weierstrass ℘- you generate field extensions from these. So when I say function, in terms of a special elliptic function. But what generate extensions, I mean that you can take roots of we want is actually a completely different uniformization those points on the elliptic curve. Just like taking the 푛th of these elliptic curves which captures the fact that the 푎 root of 5 or the cube root of 2. You can do the same and 푏 are rational numbers. It is a parametrization just thing on an elliptic curve; you can take the 푛th root of a for the rational elliptic curves. And because it captures point. These are all points which added to themselves 푛 the fact that it is defined over the rationals, it gives you times give you the point you started with. They generate a much better hold on solutions over the rationals than certain extensions of the number field you started with, the elliptic functions do. The latter really only sees the in our case the rational number field Q. complex structure. You can put a lot of restrictions on those exten- And the place it comes from are modular forms or sions. And the Selmer group is basically the smallest modular curves. To describe modular functions first: we set of extensions you can get putting on all the obvious are used to functions which satisfy the relation of being restrictions. invariant under translation. Every time we write down Let me summarize this. You’ve got the group of points. a Fourier series, we have a function which is invariant under translation. Modular functions are ones which are They generate some extensions; that’s too big—you don’t invariant under the action of a much bigger group, usually want all extensions. You cut that down as much as you can a subgroup of 푆퐿 (ℤ). So, you would ask for a function 푓(푧) using local criteria, using 푝-adic numbers; that’s called 2 in one complex variable, usually on the upper-half-plane, the Selmer group. And the essential difference between which satisfies 푓(푧) is the same as 푓((푎푧 + 푏)/(푐푧 + 푑)) the group generated by the points and the Selmer group (or, more generally, is that times a power of 푐푧 + 푑). is the Tate–Shafarevich group. So the Tate–Shafarevich These are called group gives you the error term, if you like, in trying to modular functions, get at the points via the Selmer group. Modular functions and they were ex- Raussen and Skau : Selmer’s paper, which Cassels tensively studied in refers to, studied the Diophantine equation 3푥3 + 4푦3 + hold the key to the the nineteenth century. 5푧3 = 0 and similar ones. Selmer showed that it has just Surprisingly, they hold a trivial solution in the integers, while modulo 푛 it has arithmetic of the key to the arith- nontrivial solutions for all 푛. In particular, this curve has metic of elliptic curves. no rational points. Why did Cassels invoke Selmer’s name elliptic curves. Perhaps the simplest in naming the group? way to describe it is Wiles: Yes, there are quite subtle relationships between that because we have an action of 푆퐿2(ℤ) on the upper- these. What happens is you are actually looking at one half-plane 퐻—by the action z goes to (푎푧+푏)/(푐푧+푑)—we elliptic curve, which in this case would be 푥3 +푦3 +60푧3 = can look at the quotient 퐻 modulo this action. You can 0. That is an elliptic curve, in disguise, if you like, and the then give the quotient the structure of a curve. In fact, it naturally gets the structure of a curve over the rational Tate–Shafarevich group involves looking at other ones numbers. If you take a subgroup of 푆퐿 (ℤ), or more like it, for example, 3푥3 + 4푦3 + 5푧3 = 0, which is a genus 2 precisely what is called a congruence subgroup, defined one curve but which has no rational points. Its Jacobian by the 푐 value being divisible by 푁, then you call the curve is the original elliptic curve 푥3 + 푦3 + 60푧3 = 0. One way a modular curve of level 푁. The Modularity Conjecture of describing the Tate–Shafarevich group is in terms of asserts that every elliptic curve over the rationals is actu- these curves that have genus one but don’t have rational ally a quotient of one of these modular curves for some points. And by assembling these together you can make integer 푁. It gives you a uniformization of elliptic curves the Tate–Shafarevich group, and that is reflected in the by these other entities, these modular curves. On the face Selmer group. It is too intricate to explain in words, but of it, it might seem we are losing because this is a high it is another point of view. I gave it in more arithmetic genus curve—it is more complicated. But it actually has a terminology in terms of extensions. The more geometric lot more structure because it is a moduli space. terminology was in terms of these twisted forms. Raussen and Skau : And that is a very powerful tool?

202 Notices of the AMS Volume 64, Number 3 Wiles: That is a very powerful tool, yes. You have But then, when your result is scrutinized by six referees function theory, you have deformation theory, geometric for a highly prestigious journal, it turns out that there is methods, etc. You have a lot of tools to study it. a subtle gap in one of your arguments and you are sent Raussen and Skau : Taniyama, the young Japanese back to the drawing board. After a while, you send for mathematician who first conjectured or suggested these your former student, Richard Taylor, to come to Princeton connections; his conjecture was more vague, right? to help you in your efforts. It takes a further ten months Wiles: His conjecture was more vague. He didn’t pin it of hard and frustrating work; we think we do not exagger- down to a function invariant under the modular group. I’ve ate by calling it a heroic effort under enormous pressure. forgotten exactly what he conjectured; it was invariant Then, in a sudden flash of insight, you realize that you can under some kind of group, but I forget exactly which combine some of your previous attempts with new results group he was predicting. But it was not as precise as the to circumvent the problem that had caused the gap. This congruence subgroups of the modular group. I think it turns out to be what you need in order to get the part of was originally written in Japanese, so it was not circulated the Modularity Conjecture that implies Fermat’s Last The- as widely as it might have been. I believe it was part of orem. What a relief that must have been! Would you like to notes compiled after a conference in Japan. give a few comments on this dramatic story? Raussen and Skau : It was an incredibly audacious con- Wiles: With regard jecture at that time, wasn’t it? to my own work, when Wiles: Apparently, yes. I became a profes- It is irresponsible Raussen and Skau : But then it gradually caught the at- sional mathematician tention of other mathematicians. You told us already about working with Coates, I to work on one Gerhard Frey, who came up with a conjecture relating realized I really had to Fermat’s Last Theorem with the Modularity Conjecture. stop working on Fer- problem to the Wiles: That’s right. Gerhard Frey showed that if you mat because it was take a solution to the Fermat problem, say 푎푝 + 푏푝 = 푐푝, time-consuming and I exclusion of and you create the elliptic curve 푦2 = 푥(푥 − 푎푝)(푥 + 푏푝), could see that in the everything else. then the discriminant of that curve would end up being last hundred years al- a perfect 푝th power. And if you think about what that most nothing had been means assuming the Modularity Conjecture—you have to done. And I saw others, even very distinguished mathe- assume something a bit stronger as well (the so-called maticians, had come to grief on it. When Frey came out epsilon conjecture of Serre)—then it forces this elliptic with this result, I was a bit skeptical that the Serre part curve to have the level 푁 that I spoke about to be equal of the conjecture was going to be true, but when Ribet to one, and hence the associated congruence subgroup is proved it, then, okay, this was it! equal to 푆퐿2(ℤ). But 퐻 modulo 푆퐿2(ℤ) is a curve of genus And it was a long, hard struggle. In some sense, it is zero. It has no elliptic curve quotient, so it wasn’t there irresponsible to work on one problem to the exclusion of after all, and hence there can’t be a solution to the Fermat everything else, but this is the way I tend to do things. problem. Whereas Fermat is very narrow (I mean, it is just this one equation, whose solution may or may not help with The Quest for a Proof anything else), the setting of the Modular Conjecture was Raussen and Skau : That was the point of departure for one of the big problems in number theory. It was a great your own work, with crucial further ingredients due to thing to work on anyway, so it was just a tremendous Serre and Ribet making this connection clear. May we opportunity. briefly summarize the story that then followed? It has When you are working on something like this, it takes been told by you many times, and it is the focus of a BBC many years to really build up the intuition to see the kinds documentary. of things you need and the kinds of things a solution will You had moved to the United States, first to Harvard, depend on. It’s something like discarding everything you then to Princeton University, becoming a professor there. can’t use and won’t work till your mind is so focused that When you heard of Ribet’s result, you devoted all your even making a mistake, you’ve seen enough that you’ll research time to proving the Modularity Conjecture for find another way to the end. semistable elliptic curves over the rationals. This work Funnily enough, concerning the mistake in the argu- went on for seven years of really hard work in isolation. ment that I originally gave, people have worked on that At the same time you were working as a professor in aspect of the argument and quite recently they have Princeton and you were raising small children. actually shown that you can produce arguments very like A proof seems to be accomplished in 1993, and the de- that. In fact, in every neighboring case, arguments similar velopment culminates in a series of three talks at the Isaac to the original method seem to work, but there is this Newton Institute in Cambridge back in England, announc- unique case that it doesn’t seem to work for, and there ing your proof of Fermat’s Last Theorem. You are cele- is not yet any real explanation for it. So the same kind of brated by your mathematical peers. Even the world press argument I was trying to use, using Euler systems and so takes an interest in your results, which happens very rarely on, has been made to work in every surrounding case, but for mathematical results. not the one I needed for Fermat. It’s really extraordinary.

March 2017 Notices of the AMS 203 Raussen and Skau : You once likened this quest for the Parallels to Abel’s Work proof of the Modularity Theorem to a journey through a Raussen and Skau : We want to read you a quotation: “The dark, unexplored mansion. Could you elaborate? ramparts are raised all around but, enclosed in its last re- Wiles: I started off really in the dark. I had no prior doubt, the problem defends itself desperately. Who will be insights of how the Modularity Conjecture might work or the fortunate genius who will lead the assault upon it or how you might approach it. One of the troubles with this force it to capitulate?” problem—it’s a little like the Riemann Hypothesis but Wiles: It must have been E. T. Bell, I suppose. Is it? perhaps even more so—was that you didn’t even know Raussen and Skau : No, it’s not. It is actually a what branch of mathematics the answer would be coming quote from the book Histoire des Mathématiques by from. Jean-Étienne Montucla [1725–1799], written in the late To start with, there are three ways of formulating eighteenth century. It is really the first book ever written the problem: one is geometric, one is arithmetic, and on the history of mathematics. The quotation refers to one is analytic. And there were analysts—I would not the solvability or unsolvability of the quintic equation understand their techniques at all well—who were trying by radicals. As you know, Abel [1802–1829] proved the unsolvability of the general quintic equation when he was to make headway on this problem. twenty-one years old. He worked in complete isolation, I think I was a little lucky because my natural instinct mathematically speaking, here in Oslo. Abel was obsessed, was with the arithmetic approach and I went straight or at least extremely attracted, to this problem. He also for the arithmetic route, but I could have been wrong. got a false start. He thought he could prove that one could The only previously known cases where the Modularity actually solve the quintic by radicals. Then he discovered Conjecture was known to hold were the cases of complex his mistake and he finally found the unsolvability proof. multiplication, and that proof is analytic, completely Well, this problem was, at that time, almost three hundred analytic. years old and very famous. If we move fast-forward two Partly out of necessity, I suppose, and partly because hundred years, the same quotation could be used about that’s what I knew, I went straight for an arithmetic the Fermat problem, which was around three hundred approach. I found it very useful to think about it in a fifty years old when you solved it. It is a very parallel story way that I had been studying in Iwasawa theory. With in many ways. Do you have any comments? John Coates, I had applied Iwasawa theory to elliptic Wiles: Yes. In some sense, I do feel that Abel, and then curves. When I went to Harvard, I learned about Barry Galois [1811–1832], were marking a transition in algebra Mazur’s work, where he had been studying the geometry from these equations, which were solvable in some very of modular curves using a lot of the modern machinery. simple way, to equations which can’t be solved by radicals. There were certain ideas and techniques I could draw on But this is an algebraic break that came with the quintic. from that. I realized after a while, I could actually use that In some ways, the whole trend in number theory now is to make a beginning—to find some kind of entry into the the transition from basically abelian and possibly solvable problem. extensions to insolvable extensions. How do we do the Raussen and Skau : Before you started on the Modu- arithmetic of insolvable extensions? larity Conjecture, you published a joint paper with Barry I believe the Modularity Conjecture was solved because Mazur, proving the main theorem of Iwasawa theory over we had moved on from this original abelian situation the rationals. Can you please tell us what Iwasawa theory to a nonabelian situation, and we were developing tools, is all about? modularity and so on which are fundamentally nonabelian Wiles: Iwasawa theory grew out of the work of Kummer tools. (I should say, though, that the proof got away mostly on cyclotomic fields and his approach to Fermat’s Last with using the solvable situation, not because it was more natural but because we have not solved the relevant Theorem. He studied the arithmetic, and in particular the problems in the general nonsolvable case.) ideal class groups, of prime cyclotomic fields. Iwasawa’s It is the same transition in number theory that he was idea was to consider the tower of cyclotomic fields making in algebra, which provides the tools for solving obtained by taking all 푝-power roots of unity at once. this equation. So I think it is very parallel. The main theorem of Iwasawa theory proves a relation Raussen and Skau : There is an ironic twist with Abel between the action of a generator of the Galois group on and the Fermat problem. When he was twenty-one years the 푝-primary class groups and the 푝-adic 퐿-functions. It is old, Abel came to Copenhagen to visit Professor Degen analogous to the construction used in the study of curves [1766–1825], who was the leading mathematician in Scan- over finite fields where the characteristic polynomial of dinavia at that time. Abel wrote a letter to his mentor in Frobenius is related to the zeta function. Oslo, Holmboe [1795–1850], stating three results about the Raussen and Skau : And these tools turned out to be Fermat equation without giving any proofs—one of them useful when you started to work on the Modularity Conjec- is not easy to prove, actually. This, of course, is just a cu- ture? riosity today. Wiles: They did; they gave me a starting point. It wasn’t But in the same letter, he gives vent to his frustration, obvious at the time, but when I thought about it for a intimating that he can’t understand why he gets an equa- while, I realized that there might be a way to start from tion of degree 푛2 and not 푛 when dividing the lemniscate there. arc in 푛 equal pieces. It was only after returning to Oslo

204 Notices of the AMS Volume 64, Number 3 that he discovered the double periodicity of the lemniscate of that? This is something the mind does for you. It is the integral and also of the general elliptic integral of the first flash of insight. kind. I remember (this is a trivial example in a non- If one thinks about it, what he did on the Fermat equa- mathematical setting) someone once showed me some tion turned out to be just a curiosity. But what he achieved script—it was some Gothic script—and I couldn’t make on elliptic functions, and implicitly on elliptic curves, head nor tail of it. I was trying to understand a few letters, turned out later to be a relevant tool for solving it. Of and I gave up. Then I came back half an hour later and I course, Abel had no idea that this would have anything to could read the whole thing. The mind somehow does this do with arithmetic. So this story tells us that mathematics for you and we don’t quite know how, but we do know sometimes develops in mysterious ways. what we have to do to set up the conditions where it will Wiles: It certainly does, yes. happen. Raussen and Skau : This is reminiscent of a story about Work Styles Abel. While in Berlin, he shared an apartment with some Raussen and Skau : May we ask for some comments about Norwegian friends who were not mathematicians. One of work styles of mathematicians in general and also about his friends said that Abel typically woke up during the your own? Freeman Dyson, a famous physicist and mathe- night, lit a candle, and wrote down ideas that he woke up matician at IAS in Princeton, said in his Einstein Lecture in with. Apparently his mind was working while asleep. 2008: “Some mathematicians are birds, others are frogs. Wiles: Yes, I do that, except I don’t feel the need to Birds fly high in the air and survey broad vistas of math- write them down when I wake up with it because I know ematics out to the horizon. They delight in concepts that I will not forget it. But if I have an idea when I am about unify our thinking and bring together diverse problems to go to sleep, I am terrified that I will not wake up with from different parts of the landscape. Frogs live in the mud that idea, so then I have to write it down. below and see only the flowers that grow nearby. They Raussen and Skau : Are you thinking in terms of for- delight in the details of particular objects and they solve mulas or in terms of geometric pictures or what? problems one at a time.” Wiles: It is not really Freeman Dyson didn’t say that birds were better than geometric. I think it is frogs or the other way around. He considered himself a patterns and I think I often feel that frog rather than a bird. it is just parallels be- When we are looking at your work, for us it seems rather tween situations I have doing difficult to decide where to place you in his classification seen elsewhere and the scheme: among the birds (those who create theories) or one I am facing now. In mathematics is among the frogs (those who solve problems). What is your a perfect world, what own perception? is it all pointing to? like being a Wiles: Well, I don’t feel like either. I’m certainly not a What are the ingredi- squirrel. bird—unifying different fields. I think of frogs as jumping ents that ought to go a lot. I think I’m very, very focused. I don’t know what into this proof? What the animal analogy is, but I think I’m not a frog in the am I not using that I still have in my pocket? Sometimes sense of enjoying the nearby landscape. I’m very, very it is just desperation. I assemble every piece of evidence concentrated on the problem I happen to work on and I I have and that’s all I’ve got. I have got to work with that am very selective. And I find it very hard to even take my and there is nothing else. mind off it enough to look at any of the flowers around, I often feel that doing mathematics is like being a so I don’t think that either of the descriptions quite fit. squirrel and there are some nuts at the top of a very tall Raussen and Skau : Based on your own experience, tree. But there are several trees and you don’t know which could you describe the interplay between hard, concen- one. What you do is that you run up one and you think, no, trated, and persevering work on the one side and, on the it does not look good on this one, and you go down and other side, these sudden flashes of insight that seemingly up another one, and you spend your whole life just going come out of nowhere, often appearing in a more relaxed up and down these trees. But you’ve only got up to thirty setting. Your mind must have worked unconsciously on feet. Now, if someone told you the rest of the trees—it’s the problem at hand, right? not in them, you have only one tree left—then you would Wiles: I think what you do is that you get to a situation just keep going until you found it. In some sense, it is where you know a theory so well, and maybe even more ruling out the wrong things—that is really crucial. And than one theory, so that you have seen every angle and if you just believe in your intuition and your intuition is tried a lot of different routes. correct and you stick with your one tree, then you will There is this tremendous amount of work in the find it. preparatory stage where you have to understand all the details and maybe some examples—that is your essential Problems in Mathematics launch pad. When you have developed all this, you let Raussen and Skau : Felix Klein [1849–1925] once said: the mind relax and then at some point—maybe when you “Mathematics develops as old results are being understood go away and do something else for a little bit—you come and illuminated by new methods and insights. Proportion- back and suddenly it is all clear. Why did you not think ally with a better and deeper understanding new problems

March 2017 Notices of the AMS 205 naturally arise.” And David Hilbert [1862–1943] stressed nowadays often depends on mathematical modelling that “problems are the lifeblood of mathematics.” Do you and optimization methods. Science and industry propose agree? challenges to the mathematical world. Wiles: I certainly agree with Hilbert, yes. Good problems In a sense, mathematics has become more applied than are the lifeblood of mathematics. I think you can see this it ever was. One may ask whether this is a problem for clearly in number theory in the second half of the pure mathematics. It appears that pure mathematics some- last century. For me personally, there is obviously the times is put to the sidelines, at least from the point of view Modularity Conjecture but also the whole Langlands of the funding agencies. Do you perceive this as a serious program and the Birch and Swinnerton-Dyer Conjecture. problem? These problems give you a very clear focus on what we Wiles: Well, I think in comparison with the past, one should try to achieve. We also have the Weil Conjectures feels that mathematicians two, three hundred years ago on curves and varieties over finite fields and the Mordell were able to handle a much broader spectrum of mathe- Conjecture and so on. matics, and a lot more of it touched applied mathematics These problems somehow concentrate the mind and than a typical pure mathematician would do nowadays. also simplify our goals in mathematics. Otherwise, we can On the other hand, that might be because we only remem- get very, very spread out and not sure what’s of value and ber the very best and most versatile mathematicians from what’s not of value. the past. Raussen and Skau : Do we have as good problems today I think it is always going to be a problem if funding as when Hilbert formulated his twenty-three problems in agencies are short-sighted. If they want to see a result 1900? in three years, then it is not going to work. It is hard to Wiles: I think so, yes. imagine a pure development and then the application all Raussen and Skau : Which one do you think is the most happening within three to five years. It is probably not important problem today? And how does the Langlands going to happen. program fit in? On the other hand, I don’t believe you can have a happily Wiles: Well, I think the Langlands program is the functioning applied maths world without the pure maths broadest spectrum of problems related to my field. I to back it up, providing the future and keeping them on think that the Riemann Hypothesis is the single greatest the straight and narrow. So it would be very foolish not to problem from the areas I understand. It is sometimes invest in pure mathematics. It is a bit like only investing hard to say exactly why that is, but I do believe that in energy resources that you can see now. You have to solving it would actually help solve some of these other invest in the future; you have to invest in fusion power problems. And then, of course, I have a very personal or solar power or these other things. You don’t just use attachment to the Birch and Swinnerton-Dyer Conjecture. up what is there and then start worrying about it when it Raussen and Skau : Intuition can lead us astray some- is gone. It is the same with mathematics; you can’t just times. For example, Hilbert thought that the Riemann Hy- use up the pure mathematics we have now and then start pothesis would be solved in his lifetime. There was another worrying about it when you need a pure result to generate problem on his list, the seventh, that he never thought your applications. would be solved in his lifetime, but which was solved by Gel- fond [1906–1968] in 1934. So our intuition can be wrong. Mathematical Awards Wiles: That is right. I’m not surprised that Hilbert Raussen and Skau : You have already won a lot of prizes felt that way. The Riemann Hypothesis has such a clear as a result of your achievements, culminating in proving statement, and we have the analogue in the function field Fermat’s Last Theorem. You have won the Rolf Schock setting. We understand why it is true there and we feel we Prize, given by the Swedish Academy; the Ostrowski Prize, ought to be able to translate it. Of course, many people which was given to you in Denmark; the Fermat Prize in have tried and failed. But I would still expect it to be France; the Wolf Prize in Israel; the Shaw Prize in Hong solved before the Birch and Swinnerton-Dyer Conjecture. Kong (the prize that has been named the Nobel Prize of the East), and the list goes on, culminating with the Abel Investing in Mathematics Prize tomorrow. May we ask you whether you enjoy these Raussen and Skau : Let’s hope we’ll find out in our life- awards and the accompanying celebrations? times! Wiles: I certainly love them, I have to say. I think they Classical mathematics has, roughly speaking, two are a celebration of mathematics. I think with something sources: one of them coming from the physical sciences like Fermat, it is something people are happy to see in and the other one from—let’s for simplicity call it num- their lifetime. I would obviously be very happy to see the ber theoretical speculations, with number theory not Riemann Hypothesis solved. It is just exciting to see how associated to applications. it finally gets resolved and just to understand the end That has changed. For example, your own field of ellip- of the story—because a lot of these stories we won’t live tic curves has been applied to cryptography and security. to see the end of. Each time we do see the end of such People are making money with elliptic curves nowadays! a story, it is something we will naturally celebrate. For On the other hand, many sciences apart from physics me, I learned about the Fermat problem from this book really take advantage and profit from mathematical of E. T. Bell and about the Wolfskehl Prize attached to thinking and mathematical results. Progress in industry it. The Wolfskehl Prize was still there—only just, I may

206 Notices of the AMS Volume 64, Number 3 say; I only had a few years left before the deadline for it I like to read and I like various kinds of literature: expired. novels, some biographies—it is fairly balanced. I don’t Raussen and Skau : This gives us the lead to talk a little have any other focused obsessions. When I was in school, about that prize. The Wolfskehl Prize was founded in 1906 I played on chess teams and bridge teams, but when by Paul Wolfskehl [1856–1906], who was a German physi- I started to do serious mathematics, I completely lost cian with an interest in mathematics. He bequeathed one interest in those. hundred thousand Reichmarks (equivalent to more than Raussen and Skau : What about music; are you fond of one million dollars in today’s money) to the first person to music? prove Fermat’s Last Theorem. The prize was, according to Wiles: I go and listen to concerts, but I am not the testament, valid until 13 September 2007 and you re- myself actively playing anything. I enjoy listening to ceived it in 1997. By then, due in part to the hyperinflation music—classical, preferably. Germany suffered after World War I, the prize money had Raussen and Skau : Are you interested in other sciences dwindled a lot. apart from mathematics? Wiles: For me, the amount of money was unimportant. Wiles: I would say somewhat. These are things I It was the sentimental feeling attached to the Wolfskehl do to relax, so I don’t like them to be too close to Prize that was important for me. mathematics. If it is something like animal behavior or astrophysics or something from a qualitative point of Graduate Students view—I certainly enjoy learning about those—likewise Raussen and Skau : You have had altogether twenty-one about what machines are capable of, and many other PhD students and you have attracted very gifted students. kinds of popular science, but I’m not going to spend my Some of them are really outstanding. One of them, Man- time learning the details of string theory. I’m too focused jul Bhargava, won the Fields Medal in 2014. It must be a to be willing to do that. Not that I would not be interested, pleasure to be advisor to such students. but this is my choice. Wiles: Yes, I don’t want to take too much credit for Raussen and Skau : We would like to thank you very it. In the case of Manjul, I suggested a problem to him, much for this wonderful interview, first of all on behalf but after that I had nothing much more to do. He was of the two of us but also on behalf of the Norwegian, the coming up with these absolutely marvelous discoveries. Danish, and the European Mathematical Societies. Thank In some sense, you get more credit if you have very gifted you so much! students, but the truth is that very gifted students don’t Wiles: Thank you very much! really require that much help. Raussen and Skau : What is the typical way for you of interacting with graduate students? Photo Credits Wiles: Well, I think the hardest thing to learn as a Photo of Sir Andrew J. Wiles with Crown Prince Haakon is graduate student is that afterwards you need to carry on courtesy of Audun Braastad. with the rest of your professional life; it’s hard to pick Photo of Sir Andrew J. Wiles with Martin Raussen, and problems. And if you just assign a problem and they do it, Christian Skau is courtesy of Erik F. Baardsen, DNVA. in some sense that hasn’t given them terribly much. Okay, they solved that problem, but the hard thing is then to have to go off and find other problems! So I prefer it if we come to a decision on the problem together. I give them some initial idea and which area of mathe- matics to look at, having not quite focused on the problem. Then, as they start working and become experts, they can Sidebar 1. Abel Prize Winners see a better way of pinning down what the right question is. And then they are part of the process of choosing 2016: Andrew Wiles the problem. I think that is a much better investment for 2015: John Forbes Nash Jr. and Louis Nirenberg their future. It doesn’t always work out that way, and 2014: Yakov Sinai sometimes the problem you give them turns out to be the 2013: Pierre Deligne right thing. But usually it is not that way and usually it’s 2012: Endre Szemerédi a process to find the right problem. 2011: John Milnor 2010: John Tate Hobbies and Interests 2009: Mikhail Leonidovich Gromov Raussen and Skau : We always end the Abel interviews by 2008: John G. Thompson and Jacques Tits asking the laureate what they enjoy doing when they are 2007: S. R. Srinivasa Varadhan not working with mathematics. What are your hobbies and interests outside mathematics? 2006: Lennart Carleson Wiles: Well, it varies at different times. When I was 2005: Peter Lax doing Fermat and being a father with young children, that 2004: Michael Atiyah and Isadore Singer combination was all-consuming. 2003: Jean-Pierre Serre

March 2017 Notices of the AMS 207 Sidebar 2. Notices articles on Wiles

July/August 1993: Wiles Proves Taniyama’s Conjecture; Fermat’s Last Theorem Follows, by Kenneth A. Ribet, math.berkeley.edu/~ribet/Articles/notices.pdf October 1993: Fermat Fest Draws a Crowd, by Allyn Jackson October 1994: Another Step Toward Fermat, by Allyn Jackson, www.ams.org/notices/199501/ rubin.pdf July 1995: The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles, by Gerd Faltings, www. ams.org/notices/199507/faltings.pdf July 1996: Wiles Receives NAS Award in Mathematics, by John Coates, www.ams.org/notices/ 199607/comm-wiles.pdf January 1997: Review of BBC’s Horizon Program, ”Fermat’s Last Theorem,” reviewed by Andrew Granville, www.ams.org/notices/199701/comm-granville.pdf March 1997: Announcement: 1997 Cole Prize, www.ams.org/notices/199703/comm-cole.pdf November 1997: Paul Wolfskehl and the Wolfskehl Prize, by Klaus Barner, www.ams.org/ notices/199710/barner.pdf November 1997: Book Review: Fermat’s Enigma by Simon Singh, reviewed by Allyn Jackson, www.ams.org/notices/199710/comm-fermat.pdf December 1999: Research News: A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, by Henri Darmon, www.ams.org/notices/199911/comm-darmon.pdf December 2001: Theater Review: Fermat’s Last Tango, reviewed by Robert Osserman, www.ams. org/notices/200111/rev-osserman.pdf September 2005: Wiles Receives 2005 Shaw Prize, by Allyn Jackson, www.ams.org/notices/ 200508/comm-shaw.pdf June/July 2016: Sir Andrew J. Wiles Awarded Abel Prize, www.ams.org/publications/ journals/notices/201606/rnoti-p608.pdf

208 Notices of the AMS Volume 64, Number 3 Andrew Wiles’s Marvelous Proof Henri Darmon

ermat famously claimed to have discovered “a truly marvelous proof” of his Last Theorem, which the margin of his copy of Diophantus’s Arithmetica was too narrow to contain. While this proof (if it ever existed) is lost to posterity, AndrewF Wiles’s marvelous proof has been public for over two decades and has now earned him the Abel Prize. According to the prize citation, Wiles merits this recogni- tion “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.” Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the mathematics of ancient Greece, simple enough to be understood It is also a centerpiece and appreciated Wiles giving his first lecture in Princeton about his by a novice of the “Langlands approach to proving the Modularity Conjecture in (like the ten- early 1994. year-old Andrew program,” the Wiles browsing of Theorem 2 of Karl Rubin’s contribution in this volume). the shelves of imposing, ambitious It is also a centerpiece of the “Langlands program,” the his local pub- imposing, ambitious edifice of results and conjectures lic library), yet edifice of results and which has come to dominate the number theorist’s view eluding the con- conjectures which of the world. This program has been described as a “grand certed efforts of unified theory” of mathematics. Taking a Norwegian per- the most brilliant has come to dominate spective, it connects the objects that occur in the works minds for well of Niels Hendrik Abel, such as elliptic curves and their over three cen- the number theorist’s associated Abelian integrals and Galois representations, turies, becoming with (frequently infinite-dimensional) linear representa- over its long his- view of the world. tions of the continuous transformation groups whose tory the object of study was pioneered by Sophus Lie. This report focuses lucrative awards on the role of Wiles’s theorem and its “marvelous proof” like the Wolfskehl Prize and, more importantly, motivating in the Langlands program in order to justify the closing a cascade of fundamental discoveries: Fermat’s method of phrase in the prize citation: how Wiles’s proof has opened infinite descent, Kummer’s theory of ideals, the ABC con- “a new era in number theory” and continues to have a jecture, Frey’s approach to ternary diophantine equations, profound and lasting impact on mathematics. Serre’s conjecture on mod 푝 Galois representations,…. Our “beginner’s tour” of the Langlands program will Even without its seemingly serendipitous connection only give a partial and undoubtedly biased glimpse of to Fermat’s Last Theorem, Wiles’s modularity theorem the full panorama, reflecting the author’s shortcomings is a fundamental statement about elliptic curves (as evi- as well as the inherent limitations of a treatment aimed denced, for instance, by the key role it plays in the proof at a general readership. We will motivate the Langlands program by starting with a discussion of diophantine Henri Darmon is James McGill Professor of Mathematics at McGill equations: for the purposes of this exposition, they are University and a member of CICMA (Centre Interuniversitaire en equations of the form Calcul Mathématique Algébrique) and CRM (Centre de Recherches Mathématiques). His e-mail address is [email protected]. (1) 풳 ∶ 푃(푥1, … , 푥푛+1) = 0,

This report is a very slightly expanded transcript of the Abel Prize where 푃 is a polynomial in the variables 푥1, … , 푥푛+1 lecture delivered by the author on May 25, 2016, at the University with integer (or sometimes rational) coefficients. One can of Oslo. examine the set, denoted 풳(퐹), of solutions of (1) with For permission to reprint this article, please contact: coordinates in any ring 퐹. As we shall see, the subject [email protected]. draws much of its fascination from the deep and subtle DOI: http://dx.doi.org/10.1090/noti1487 ways in which the behaviours of different solution sets

March 2017 Notices of the AMS 209 can resonate with each other, even if the sets 풳(ℤ) or (2) the reciprocal roots of 푄(푇) and 푅(푇) are complex 풳(ℚ) of integer and rational solutions are foremost in numbers of absolute value 푝푖/2 with 푖 an integer our minds. Examples of diophantine equations include in the interval 0 ≤ 푖 ≤ 2푛. 푑 푑 푑 Fermat’s equation 푥 + 푦 = 푧 , the Brahmagupta–Pell The first statement—the rationality of the zeta function, 2 2 equation 푥 −퐷푦 = 1 with 퐷 > 0, as well as elliptic curve which was proved by Bernard Dwork in the early 1960s— 2 3 equations of the form 푦 = 푥 + 푎푥 + 푏, in which 푎 and 푏 is a key part of the Weil conjectures, whose formulation are rational parameters, the solutions (푥, 푦) with rational in the 1940s unleashed a revolution in arithmetic ge- coordinates being the object of interest in the latter case. ometry, driving the development of étale cohomology It can be instructive to approach a diophantine equation by Grothendieck and his school. The second statement, −푠 by first studying its solutions over simpler rings, such as which asserts that the complex function 휁푝(풳; 푝 ) has the complete fields of real or complex numbers. The set its roots on the real lines ℜ(푠) = 푖/2 with 푖 as above, (2) ℤ/푛ℤ ∶= {0, 1, … , 푛 − 1} is known as the Riemann hypothesis for the zeta func- tions of diophantine equations over finite fields. It was of remainders after division by an integer 푛 ≥ 2, equipped proved by Pierre Deligne in 1974 and is one of the major with its natural laws of addition, subtraction, and mul- achievements for which he was awarded the Abel Prize in tiplication, is another particularly simple collection of 2013. numbers of finite cardinality. If 푛 = 푝 is prime, this ring That the asymptotic behaviour of 푁 can lead to deep is even a field: it comes equipped with an operation of 푝 insights into the behaviour of the associated diophantine division by nonzero elements, just like the more familiar collections of rational, real, or complex numbers. The fact equations is one of the key ideas behind the Birch and Swinnerton-Dyer conjecture. Understanding the patterns that 픽푝 ∶= ℤ/푝ℤ is a field is an algebraic characterisation of the primes that forms the basis for most known ef- satisfied by the function ficient primality tests and factorisation algorithms. One (9) 푝 ↦ 푁푝 or 푝 ↦ 휁푝(풳; 푇) of the great contributions of Evariste Galois, in addition as the prime 푝 varies will also serve as our motivating to the eponymous theory which plays such a crucial role question for the Langlands program. in Wiles’s work, is his discovery of a field of cardinality 푟 푟 It turns out to be fruitful to package the zeta functions 푝 for any prime power 푝 . This field, denoted 픽 푟 and 푝 over all the finite fields into a single function of a complex sometimes referred to as the Galois field with 푝푟 elements, variable 푠 by taking the infinite product is even unique up to isomorphism. −푠 For a diophantine equation 풳 as in (1), the most basic (10) 휁(풳; 푠) = ∏ 휁푝(풳; 푝 ) invariant of the set 푝 푛+1 as 푝 ranges over all the prime numbers. In the case of the (푥1, … , 푥푛+1) ∈ 픽푝푟 such that (3) 풳(픽푝푟 ) ∶= { } simplest nontrivial diophantine equation 푥 = 0, whose 푃(푥1, … , 푥푛+1) = 0 solution set over 픽푝푟 consists of a single point, one has 푟 of solutions over 픽푝 is of course its cardinality 푁푝푟 = 1 for all 푝, and therefore (4) 푁푝푟 ∶= #풳(픽푝푟 ). 푇푟 (11) 휁 (푥 = 0; 푇) = exp ( ) = (1 − 푇)−1. What patterns (if any) are satisfied by the sequence 푝 ∑ 푟≥1 푟 (5) 푁푝, 푁푝2 , 푁푝3 , … , 푁푝푟 ,…? It follows that This sequence can be packaged into a generating series 1 −1 like (12) 휁(푥 = 0; 푠) = ∏ (1 − ) 푝푠 ∞ ∞ 푝 푟 푁푝푟 푟 (6) ∑ 푁푝푟 푇 or ∑ 푇 . 1 1 1 푟=1 푟=1 푟 (13) = ∏ (1 + + + + ⋯) 푝푠 푝2푠 푝3푠 For technical reasons it is best to consider the exponential 푝 ∞ of the latter: 1 (14) = ∑ = 휁(푠). ∞ 푛푠 푁푝푟 푟 푛=1 (7) 휁푝(풳; 푇) ∶= exp ( ∑ 푇 ). 푟=1 푟 The zeta function of even the humblest diophantine This power series in 푇 is known as the zeta function equation is thus a central object of mathematics: the celebrated Riemann zeta function, which is tied to some of 풳 over 픽푝. It has integer coefficients and enjoys the following remarkable properties: of the deepest questions concerning the distribution of prime numbers. In his great memoir of 1860, Riemann (1) It is a rational function in 푇: proved that, even though (13) and (14) only converge 푄(푇) absolutely on the right half-plane ℜ(푠) > 1, the function (8) 휁푝(풳; 푇) = , 푅(푇) 휁(푠) extends to a meromorphic function of 푠 ∈ ℂ (with a where 푄(푇) and 푅(푇) are polynomials in 푇 whose single pole at 푠 = 1) and possesses an elegant functional degrees (for all but finitely many 푝) are inde- equation relating its values at 푠 and 1 − 푠. The zeta pendent of 푝 and determined by the shape—the functions of linear equations 풳 in 푛 + 1 variables are just complex topology—of the set 풳(ℂ) of complex shifts of the Riemann zeta function, since 푁푝푟 is equal to solutions; 푝푛푟, and therefore 휁(풳; 푠) = 휁(푠 − 푛).

210 Notices of the AMS Volume 64, Number 3 Moving on to equations of degree two, the general involves the same basic constituents, Dirichlet series, quadratic equation in one variable is of the form 푎푥2 +푏푥+ as in the one-variable case. This means that quadratic 푐 = 0, and its behaviour is governed by its discriminant diophantine equations in any number of variables are well understood, at least as far as their zeta functions are (15) Δ ∶= 푏2 − 4푎푐. concerned. This purely algebraic fact remains true over the finite The plot thickens when equations of higher degree fields, and for primes 푝 ∤ 2푎Δ one has are considered. Consider for instance the cubic equation 3 0 if Δ is a nonsquare modulo 푝, 푥 − 푥 − 1 of discriminant Δ = −23. For all 푝 ≠ 23, (16) 푁 = { 푝 2 if Δ is a square modulo 푝. this cubic equation has no multiple roots over 픽푝푟 , and therefore 푁푝 = 0, 1, or 3. A simple expression for 푁푝 in A priori, the criterion for whether 푁푝 = 2 or 0—whether this case is given by the following theorem of Hecke: the integer Δ is or is not a quadratic residue modulo 푝—seems like a subtle condition on the prime 푝. To get Theorem (Hecke). The following hold for all primes 푝 ≠ a better feeling for this condition, consider the example 23: 2 of the equation 푥 − 푥 − 1, for which Δ = 5. Calculating (1) If 푝 is not a square modulo 23, then 푁푝 = 1. whether 5 is a square or not modulo 푝 for the first few (2) If 푝 is a square modulo 23, then primes 푝 ≤ 101 leads to the following list: 0 if 푝 = 2푎2 + 푎푏 + 3푏2, (17) (21) 푁 = { 푝 3 if 푝 = 푎2 + 푎푏 + 6푏2, 2 for 푝=11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, … 푁 ={ 푝 0 for 푝=2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, … for some 푎, 푏 ∈ ℤ. A clear pattern emerges from this experiment: whether Hecke’s theorem implies that 푁 = 0 or 2 seems to depend only on the rightmost digit ∞ 푝 3 −푠 of 푝, i.e., on what the remainder of 푝 is modulo 10. One (22) 휁(푥 − 푥 − 1; 푠) = 휁(푠) × ∑ 푎푛푛 , is led to surmise that 푛=1 where the generating series 2 if 푝 ≡ 1, 4 (mod 5), (18) 푁푝 = { (23) 0 if 푝 ≡ 2, 3 (mod 5), 푛 2 3 6 8 13 16 23 퐹(푞)∶=∑ 푎푛푞 = 푞−푞 −푞 +푞 +푞 −푞 −푞 +푞 +⋯ a formula that represents a dramatic improvement over is given by the explicit formula (16), allowing a much more efficient calculation of 푁푝 for example. The guess in (18) is in fact a consequence of 1 2 2 2 2 (24) 퐹(푞) = ⎛ ∑ 푞푎 +푎푏+6푏 − 푞2푎 +푎푏+3푏 ⎞ . Gauss’s celebrated law of quadratic reciprocity: 2 ⎝푎,푏∈ℤ ⎠ 2 Theorem (Quadratic reciprocity). For any equation 푎푥 + The function 푓(푧) = 퐹(푒2휋푖푧) that arises by setting 푞 = 2 푏푥+푐, with Δ ∶= 푏 −4푎푐, the value of the function 푝 ↦ 푁푝 푒2휋푖푧 in (24) is a prototypical example of a modular form: (for 푝 ∤ 푎Δ) depends only on the residue class of 푝 modulo namely, it satisfies the transformation rule 4Δ and hence is periodic with period length dividing 4|Δ|. (25) 푎푧 + 푏 푎, 푏, 푐, 푑 ∈ ℤ, 푎푑 − 푏푐 = 1 The repeating pattern satisfied by the 푁푝’s as 푝 varies 푓 ( ) = (푐푧+푑)푓(푧), { 푎 greatly facilitates the manipulation of the zeta functions 푐푧 + 푑 23|푐, ( 23 ) = 1. of quadratic equations. For example, the zeta function of under so-called modular substitutions of the form 푧 ↦ 푎푧+푏 . 풳 ∶ 푥2 − 푥 − 1 = 0 is equal to 푐푧+푑 This property follows from the Poisson summation for- 1 1 1 1 1 1 1 휁(풳; 푠) = 휁(푠)×{(1− − + )+( − − + ) mula applied to the expression in (24). Thanks to (25), the 2푠 3푠 4푠 6푠 7푠 8푠 9푠 zeta function of 풳 can be manipulated with the same ease 1 1 1 1 as the zeta functions of Riemann and Dirichlet. Indeed, (19) + (1 − − + )+⋯}. 푠 푠 푠 푠 ∞ −푠 11 12 13 14 Hecke showed that the 퐿-series ∑푛=1 푎푛푛 attached to ∞ 2휋푖푛푧 The series that occurs on the right-hand side is a prototyp- a modular form ∑푛=1 푎푛푒 possesses very similar an- ical example of a Dirichlet 퐿-series. These 퐿-series, which alytic properties, notably an analytic continuation and a are the key actors in the proof of Dirichlet’s theorem on Riemann-style functional equation. the infinitude of primes in arithmetic progressions, enjoy The generating series 퐹(푞) can also be expressed as an many of the same analytic properties as the Riemann zeta infinite product: function: an analytic continuation to the entire complex (26) ∞ plane and a functional equation relating their values at 푠 1 2 2 2 2 ⎛ 푞푎 +푎푏+6푏 −푞2푎 +푎푏+3푏 ⎞=푞 (1−푞푛)(1−푞23푛). and 1 − 푠. They are also expected to satisfy a Riemann hy- ∑ ∏ 2 ⎝푎,푏∈ℤ ⎠ 푛=1 pothesis which generalises Riemann’s original statement and is just as deep and elusive. The first few terms of this power series identity can readily It is a (not completely trivial) fact that the zeta function be verified numerically, but its proof is highly nonobvious of the general quadratic equation in 푛 variables and indirect. It exploits the circumstance that the space of holomorphic functions of 푧 satisfying the transformation 푛 푛 rules (25) together with suitable growth properties is a one- (20) ∑ 푎푖푗푥푖푥푗 + ∑ 푏푖푥푖 + 푐 = 0 푖,푗=1 푖=1 dimensional complex vector space which also contains

March 2017 Notices of the AMS 211 the infinite product above. Indeed, the latter is equal to after a suitable change of variables and which are non- 휂(푞)휂(푞23), where singular, a condition equivalent to the assertion that the 3 2 ∞ discriminant Δ ∶= −16(4푎 + 27푏 ) is nonzero. (27) 휂(푞) = 푞1/24 ∏(1 − 푞푛) To illustrate Wiles’s theorem with a concrete example, 푛=1 consider the equation is the Dedekind eta function whose logarithmic derivative (32) 퐸 ∶ 푦2 = 푥3 − 푥, (after viewing 휂 as a function of 푧 through the change of 2휋푖푧 of discriminant Δ = 64. After setting variables 푞 = 푒 ) is given by (33) ∞ −푠 −푠 −푠 −1 휂′(푧) −1 휁(퐸; 푠) = 휁(푠−1)×(푎1 + 푎22 + 푎33 + 푎44 + ⋯) , (28) = −휋푖( + 2 ∑ ( ∑ 푑)푒2휋푖푛푧) 휂(푧) 12 푛=1 푑|푛 the associated generating series satisfies the following identities reminiscent of (24) and (26): 푖 ∞ ∞ 1 (29) = , (34) ∑ ∑ 2 4휋 푚=−∞ 푛=−∞ (푚푧 + 푛) 푛 5 9 13 17 25 where the term attached to (푚, 푛) = (0, 0) is excluded 퐹(푞) = ∑ 푎푛푞 = 푞 − 2푞 − 3푞 + 6푞 + 2푞 − 푞 + ⋯ 2 2 from the last sum. The Dedekind 휂-function is also (35) = ∑ 푎 ⋅ 푞(푎 +푏 ) connected to the generating series for the partition 푎,푏 function 푝(푛) describing the number of ways in which ∞ 푛 can be expressed as a sum of positive integers via the (36) = 푞 ∏(1 − 푞4푛)2(1 − 푞8푛)2, identity 푛=1 2 ∞ where the sum in (35) runs over the (푎, 푏) ∈ ℤ for which (30) 휂−1(푞) = 푞−1/24 ∑ 푝(푛)푞푛, the Gaussian integer 푎 + 푏푖 is congruent to 1 modulo 푛=0 (1+푖)3. (This identity follows from Deuring’s study of zeta which plays a starring functions of elliptic curves with complex multiplication role alongside Jeremy and may even have been known earlier.) Once again, the 2휋푖푧 “There are five Irons and Dev Patel in holomorphic function 푓(푧) ∶= 퐹(푒 ) is a modular form a recent film about satisfying the slightly different transformation rule elementary the life of Srinivasa (37) 푎푧 + 푏 푎, 푏, 푐, 푑 ∈ ℤ, 푎푑 − 푏푐 = 1, Ramanujan. 푓 ( ) = (푐푧+푑)2푓(푧), { arithmetical Commenting on the 푐푧 + 푑 32|푐. operations: “unreasonable effec- Note the exponent 2 that appears in this formula. Because tiveness and ubiquity of it, the function 푓(푧) is said to be a modular form of addition, of modular forms,” weight 2 and level 32. The modular forms of (25) attached Martin Eichler once to cubic equations in one variable are of weight 1, but subtraction, wrote, “There are otherwise the parallel of (35) and (36) with (24) and (26) five elementary arith- is striking. The original conjecture of Shimura–Taniyama, multiplication, metical operations: and Weil asserts the same pattern for all elliptic curves: addition, subtraction, division,…and multiplication, divi- Conjecture (Shimura–Taniyama–Weil). Let 퐸 be any ellip- sion,…and modular tic curve. Then modular forms.” ∞ −1 forms.” Equations (26), (38) 휁(퐸; 푠) = 휁(푠 − 1) × ( 푎 푛−푠) , (29), and (30) are just a ∑ 푛 푛=1 few of the many won- 2휋푖푛푧 drous identities which abound, like exotic strains of where 푓퐸(푧) ∶= ∑ 푎푛푒 is a modular form of weight 2. fragrant wild orchids, in what Roger Godement has called The conjecture was actually more precise and predicted the “garden of modular delights.” that the level of 푓퐸—i.e., the integer that appears in the The example above and many others of a similar type transformation property for 푓퐸 as the integers 23 and 32 are described in Jean-Pierre Serre’s delightful monograph in (25) and (37) respectively—is equal to the arithmetic [Se], touching on themes that were also covered in Serre’s conductor of 퐸. This conductor, which is divisible only by lecture at the inaugural Abel Prize ceremony in 2003. primes for which the equation defining 퐸 becomes singular Hecke was able to establish that all cubic polynomials in modulo 푝, is a measure of the arithmetic complexity of one variable are modular; i.e., the coefficients of their zeta 퐸 and can be calculated explicitly from an equation for functions obey patterns just like those of (24) and (25). 퐸 by an algorithm of Tate. An elliptic curve is said to be Wiles’s achievement was to extend this result to a large semistable if its arithmetic conductor is squarefree. This class of cubic diophantine equations in two variables over class of elliptic curves includes those of the form the rational numbers: the elliptic curve equations which (39) 푦2 = 푥(푥 − 푎)(푥 − 푏) can be brought to the form with gcd(푎, 푏) = 1 and 16|푏. The most famous elliptic (31) 푦2 = 푥3 + 푎푥 + 푏 curves in this class are those that ultimately do not exist:

212 Notices of the AMS Volume 64, Number 3 compact subgroup GL2(ℤ푝) for all but finitely many 푝). The shift in emphasis from modular forms to the so-called automorphic representations which they span is decisive. Langlands showed how to attach an 퐿-function to any irreducible automorphic representation of 퐺(픸ℚ) for an arbitrary reductive algebraic group 퐺, of which the matrix groups GL푛 and more general algebraic groups of Lie type are prototypical examples. This greatly enlarges the notion of what it means to be “modular”: a diophantine equation is now said to have this property if its zeta function can be expressed in terms of the Langlands 퐿- functions attached to automorphic representations. One of the fundamental goals in the Langlands program is to establish further cases of the following conjecture: Conjecture. All diophantine equations are modular in the above sense. This conjecture can be viewed as a far-reaching gen- eralisation of quadratic reciprocity and underlies the non-Abelian reciprocity laws that are at the heart of Andrew Wiles’s achievement. Before Wiles’s proof, the following general classes of diophantine equations were known to be modular: Andrew Wiles, Henri Darmon, and Mirela Çiperiani in • Quadratic equations, by Gauss’s law of quadratic June 2016 at Harvard University during a conference reciprocity; in honor of Karl Rubin’s sixtieth birthday. • Cubic equations in one variable, by the work of Hecke and Maass; the “Frey curves” 푦2 = 푥(푥 − 푎푝)(푥 + 푏푝) arising from • Quartic equations in one variable. 푝 푝 푝 putative solutions to Fermat’s equation 푎 + 푏 = 푐 , This last case deserves further comment, since it has whose nonexistence had previously been established in a not been discussed previously and plays a primordial 1 landmark article of Kenneth Ribet under the assumption role in Wiles’s proof. The modularity of quartic equations of their modularity. It is the proof of the Shimura– follows from the seminal work of Langlands and Tunnell. Taniyama–Weil conjecture for semistable elliptic curves While it is beyond the scope of this survey to describe that earned Andrew Wiles the Abel Prize: their methods, it must be emphasised that Langlands and Theorem (Wiles). Let 퐸 be a semistable elliptic curve. Then Tunnell make essential use of the solvability by radicals of the general quartic equation, whose underlying symmetry 퐸 satisfies the Shimura–Taniyama–Weil conjecture. group is contained in the permutation group 푆4 on 4 letters. The semistability assumption in Wiles’s theorem was Solvable extensions are obtained from a succession of later removed by Christophe Breuil, Brian Conrad, Fred Abelian extensions, which fall within the purview of the Diamond, and Richard Taylor around 1999. (See, for class field theory developed in the nineteenth and first instance, the account [Da] that appeared in the Notices at half of the twentieth century. On the other hand, the the time.) modularity of the general equation of degree > 4 in one As a prelude to describing some of the important ideas variable, which cannot be solved by radicals, seemed to in its proof, one must first try to explain why Wiles’s lie well beyond the scope of the techniques that were theorem occupies such a central position in mathematics. available in the “pre-Wiles era.” The reader who perseveres The Langlands program places it in a larger context to the end of this essay will be given a glimpse of how by offering a vast generalisation of what it means for our knowledge of the modularity of the general quintic a diophantine equation to be “associated to a modular equation has progressed dramatically in the wake of form.” The key is to view modular forms attached to cubic Wiles’s breakthrough. equations or to elliptic curves as in (24) or (34) as vectors Prior to Wiles’s proof, modularity was also not known in certain irreducible infinite-dimensional representations for any interesting general class of equations (of degree of the locally compact topological group > 2, say) in more than one variable; in particular it had only been verified for finitely many elliptic curves over ℚ (40) GL (픸 ) = ′ GL (ℚ ) × GL (ℝ) 2 ℚ ∏ 푝 2 푝 2 up to isomorphism over ℚ̄ (including the elliptic curves ′ (where ∏푝 denotes a restricted direct product over all over ℚ with complex multiplication, of which the elliptic the prime numbers consisting of elements (훾푝)푝 for curve of (31) is an instance.) Wiles’s modularity theorem which the 푝th component 훾푝 belongs to the maximal confirmed the Langlands conjectures in the important test case of elliptic curves, which may seem like (and, in 1See the interview with Ribet as the new AMS president in this fact, are) very special diophantine equations, but have issue, page 229. provided a fertile terrain for arithmetic investigations,

March 2017 Notices of the AMS 213 both in theory and in applications (cryptography, coding Like diophantine equations, Galois representations also theory…). give rise to analogous zeta functions. More precisely, the Returning to the main group 퐺ℚ,푆 contains, for each prime 푝 ∉ 푆, a distinguished theme of this report, element called the Frobenius element at 푝, denoted 휎푝. Wiles’s proof is Wiles’s proof is also impor- Strictly speaking, this element is well defined only up tant for having introduced to conjugacy in 퐺ℚ,푆, but this is enough to make the also important a revolutionary new ap- arithmetic sequence proach which has opened 푟 (42) 푁푝푟 (휚) ∶= Trace(휚(휎푝 )) for having the floodgates for many well defined. The zeta function 휁(휚; 푠) packages the further breakthroughs in information from this sequence in exactly the same way introduced a the Langlands program. as in the definition of 휁(풳; 푠). To expand on this revolutionary For example, if 풳 is attached to a polynomial 푃 of point, we need to present degree 푑 in one variable, the action of 퐺 on the another of the dramatis ℚ,푆 new approach roots of 푃 gives rise to a 푑-dimensional permutation personae in Wiles’s proof: representation which has Galois representations. Let ̄ (43) 휚 ∶ 퐺 ⟶ GL (ℚ), opened the 퐺ℚ = Gal(ℚ/ℚ) be the 풳 ℚ,푆 푑 absolute Galois group of and 휁(풳, 푠) = 휁(휚풳, 푠). This connection goes far deeper, floodgates for ℚ, namely, the automor- extending to diophantine equations in 푛 + 1 variables for phism group of the field general 푛 ≥ 0. The glorious insight at the origin of the many further of all algebraic numbers. Weil conjectures is that 휁(풳; 푠) can be expressed in terms It is a profinite group, of the zeta functions of Galois representations arising breakthroughs endowed with a natu- in the étale cohomology of 풳, a cohomology theory with ral topology for which ℓ-adic coefficients which associates to 풳 a collection in the Langlands the subgroups Gal(ℚ/퐿)̄ 푖 {퐻et(풳/ℚ̄ ,ℚℓ)} with 퐿 ranging over the 0≤푖≤2푛 program. finite extensions of ℚ of finite-dimensional ℚℓ-vector spaces endowed with a form a basis of open continuous linear action of 퐺ℚ,푆. (Here 푆 is the set of subgroups. Following the primes 푞 consisting of ℓ and the primes for which the original point of view taken by Galois himself, the group equation of 풳 becomes singular after being reduced 퐺ℚ acts naturally as permutations on the roots of poly- modulo 푞.) These representations generalise the repre- nomials with rational coefficients. Given a finite set 푆 of sentation 휚풳 of (43), insofar as the latter is realised by 0 primes, one may consider only the monic polynomials the action of 퐺ℚ,푆 on 퐻et(풳ℚ̄ ,ℚℓ) after extending the with integer coefficients whose discriminant is divisible coefficients from ℚ to ℚℓ. only by primes ℓ ∈ 푆 (eventually after a change of vari- Theorem (Weil, Grothendieck,…). If 풳 is a diophantine ables). The topological group 퐺ℚ operates on the roots of equation having good reduction outside 푆, there exist Ga- such polynomials through a quotient, denoted 퐺ℚ,푆—the lois representations 휚1 and 휚2 of 퐺ℚ,푆 for which automorphism group of the maximal algebraic extension unramified outside 푆, which can be regarded as the sym- (44) 휁(풳; 푠) = 휁(휚1; 푠)/휁(휚2; 푠). 푖 metry group of all the zero-dimensional varieties over ℚ The representations 휚1 and 휚2 occur in ⊕퐻et(풳/ℚ̄ ,ℚℓ), having “nonsingular reduction outside 푆.” where the direct sum ranges over the odd and even values In addition to the permutation representations of 퐺ℚ of 0 ≤ 푖 ≤ 2푛 for 휚1 and 휚2 respectively. More canonically, that were so essential in Galois’s original formulation there are always irreducible representations 휚1, … , 휚푡 of of his theory, it has become important to study the 퐺ℚ,푠 and integers 푑1, … 푑푡 such that (continuous) linear representations 푡 푑푖 (45) 휁(풳; 푠) = ∏ 휁(휚푖; 푠) , (41) 휚 ∶ 퐺ℚ,푆 ⟶ 퐺퐿푛(퐿) 푖=1 of this Galois group, where 퐿 is a complete field, such as arising from the decompositions of the (semisimplifi- the fields ℝ or ℂ of real or complex numbers, the finite 푖 cation of the) 퐻et(풳ℚ̄ ,ℚℓ) into a sum of irreducible field 픽ℓ푟 equipped with the discrete topology, or a finite representations. The 휁(휚푖, 푠) can be viewed as the ̄ extension 퐿 ⊂ ℚℓ of the field ℚℓ of ℓ-adic numbers. “atomic constituents” of 휁(풳, 푠) and reveal much of Galois representations were an important theme in the the “hidden structure” in the underlying equation. The de- work of Abel and remain central in modern times. Many composition of 휁(풳; 푠) into a product of different 휁(휚푖; 푠) illustrious mathematicians in the twentieth century have is not unlike the decomposition of a wave function into contributed to their study, including three former Abel its simple harmonics. Prize winners: Jean-Pierre Serre, John Tate, and Pierre A Galois representation is said to be modular if its zeta Deligne. Working on Galois representations might seem function can be expressed in terms of generating series to be a prerequisite for an algebraic number theorist to attached to modular forms and automorphic representa- receive the Abel Prize! tions and is said to be geometric if it can be realised in

214 Notices of the AMS Volume 64, Number 3 an étale cohomology group of a diophantine equation as the inverse limit being taken relative to the multiplication- 푛 above. The “main conjecture of the Langlands program” by-3 maps. The groups 퐸[3 ] and 푇3(퐸) are free modules 푛 can now be amended as follows: of rank 2 over (ℤ/3 ℤ) and ℤ3 respectively and are endowed with continuous linear actions of 퐺 , where 푆 Conjecture. All geometric Galois representations of 퐺 ℚ,푆 ℚ,푆 is a set of primes containing 3 and the primes that divide are modular. the conductor of 퐸. One obtains the associated Galois Given a Galois representation representations:

(46) 휚 ∶ 퐺ℚ,푆 ⟶ GL푛(ℤ ) 휚퐸,3̄ ∶ 퐺ℚ,푆 ⟶ Aut(퐸[3]) ≃ GL2(픽3), ℓ (50) with ℓ-adic coefficients, one may consider the resulting 휚퐸,3 ∶ 퐺ℚ,푆 ⟶ GL2(ℤ3). mod ℓ representation The theorem of Langlands and Tunnell about the modular-

(47) 휚̄ ∶ 퐺ℚ,푆 ⟶ GL푛(픽ℓ). ity of the general quartic equation leads to the conclusion that 휚̄ is modular. This rests on the happy circumstance The passage from 휚 to 휚̄ amounts to replacing the 퐸,3 푟 that quantities 푁푝푟 (휚) ∈ ℤℓ as 푝 ranges over all the prime powers with their (51) GL2(픽3)/⟨±1⟩ ≃ 푆4, mod ℓ reduction. and hence that 퐸[3] has essentially the same symmetry Such a passage would Since then, group as the general quartic equation! The isomorphism seem rather contrived in (51) can be realised by considering the action of GL2(픽3) for the sequences “modularity lifting on the set {0, 1, 2, ∞} of points on the projective line over 푁 푟 (풳)—why study 푝 픽3. the solution counts of theorems” have If 퐸 is semistable, Wiles is able to check that both a diophantine equa- 휚퐸,3 and 휚퐸,3̄ satisfy the conditions necessary to apply tion over different proliferated, and the Modularity Lifting Theorem, at least when 휚퐸,3̄ is finite fields, taken irreducible. It then follows that 휚퐸,3 is modular, and modulo ℓ?—if one their study, in ever therefore so is 퐸, since 휁(퐸; 푠) and 휁(휚퐸,3; 푠) are the same. did not know a pri- more general and Note the key role played by the result of Langlands– ori that these counts Tunnell in the above strategy. It is a dramatic illustration arise from ℓ-adic Ga- delicate settings, fo the unity and historical continuity of mathematics that lois representations the solution in radicals of the general quartic equation, with coefficients in ℤℓ. has spawned an one of the great feats of the algebraists of the Italian There is a correspond- industry. Renaissance, is precisely what allowed Langlands, Tunnell, ing notion of what it and Wiles to prove their modularity results more than means for 휚̄ to be mod- five centuries later. ular, namely, that the Having established the modularity of all semistable el- data of 푁 푟 (휚)̄ agrees, very loosely speaking, with the 푝 liptic curves 퐸 for which 휚퐸,3̄ is irreducible, Wiles disposes mod ℓ reduction of similar data arising from an automor- of the others by applying his lifting theorem to the prime phic representation. We can now state Wiles’s celebrated ℓ = 5 instead of ℓ = 3. The Galois representation 휚퐸,5̄ is modularity lifting theorem, which lies at the heart of his always irreducible in this setting, because no elliptic curve strategy: over ℚ can have a rational subgroup of order 15. Nonethe- Wiles’s Modularity Lifting Theorem. Let less, the approach of exploiting ℓ = 5 seems hopeless at first glance, because the Galois representation 퐸[5] is (48) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℤℓ) not known to be modular a priori, for much the same be an irreducible geometric Galois representation satisfy- reason that the general quintic equation cannot be solved ing a few technical conditions (involving, for the most part, by radicals. (Indeed, the symmetry group SL2(픽5) is a ̄ double cover of the alternating group 퐴 on 5 letters and the restriction of 휚 to the subgroup 퐺ℚℓ = Gal(ℚℓ/ℚℓ) of 5 퐺ℚ,푆). If 휚̄ is modular and irreducible, then so is 휚. thus is closely related to the symmetry group underlying the general quintic.) To establish the modularity of 퐸[5], This stunning result was completely new at the time: Wiles constructs an auxiliary semistable elliptic curve 퐸′ nothing remotely like it had ever been proved before! Since satisfying then, “modularity lifting theorems” have proliferated, and their study, in ever more general and delicate settings, has (52) 휚퐸̄ ′,5 = 휚퐸,5̄ , 휚퐸̄ ′,3 is irreducible. spawned an industry and led to a plethora of fundamental It then follows from the argument in the previous para- advances in the Langlands program. graph that 퐸′ is modular, hence that 퐸′[5] = 퐸[5] is Let us first explain how Wiles himself parlays his modular as well, putting 퐸 within striking range of the original modularity lifting theorem into a proof of the modularity lifting theorem with ℓ = 5. This lovely epi- Shimura–Taniyama-Weil conjecture for semistable elliptic logue of Wiles’s proof, which came to be known as the curves. Given such an elliptic curve 퐸, consider the groups “3-5 switch,” may have been viewed as an expedient trick (49) at the time. But since then the prime switching argument 푛 푛 푛 퐸[3 ] ∶= {푃 ∈ 퐸(ℚ)̄ ∶ 3 푃 = 0} , 푇3(퐸) ∶= lim 퐸[3 ], ← has become firmly embedded in the subject, and many

March 2017 Notices of the AMS 215 variants of it have been exploited to spectacular effect in • The Sato–Tate conjecture concerning the distribution deriving new modularity results. of the numbers 푁푝(퐸) for an elliptic curve 퐸 as the Wiles’s modularity prime 푝 varies, whose proof was known to follow lifting theorem reveals from the modularity of all the symmetric power The modularity of that “modularity is Galois representations attached to 퐸, was proved in contagious” and can large part by Laurent Clozel, Michael Harris, Nick elliptic curves was often be passed on to Shepherd-Barron, and Richard Taylor around 2006. an ℓ-adic Galois rep- • One can also make sense of what it should mean only the first in a resentation from its for diophantine equations over more general number mod ℓ reduction. It fields to be modular. The modularity of elliptic curves series of is this simple prin- over all real quadratic fields has been proved very ciple that accounts recently by Nuno Freitas, Bao Le Hung, and Samir Siksek spectacular for the tremendous by combining the ever more general and powerful applications. impact that the Modu- modularity lifting theorems currently available with a larity Lifting Theorem, careful diophantine study of the elliptic curves which and the many variants could a priori fall outside the scope of these lifting proved since then, continue to have on the subject. Indeed, theorems. the modularity of elliptic curves was only the first in a • Among the spectacular recent developments building series of spectacular applications of the ideas introduced on Wiles’s ideas is the proof, by Laurent Clozel and by Wiles, and since 1994 the subject has witnessed a Jack Thorne, of the modularity of certain symmetric real golden age, in which open problems that previously powers of the Galois representations attached to seemed completely out of reach have succumbed one by holomorphic modular forms, which is described in one. Thorne’s contribution to this volume. Among these developments, let us mention: These results are just a sampling of the transformative • The two-dimensional Artin conjecture, first formu- impact of modularity lifting theorems. The Langlands pro- lated in 1923, concerns the modularity of all odd, gram remains a lively area, with many alluring mysteries two-dimensional Galois representations yet to be explored. It is hard to predict where the next breakthroughs will come, but surely they will continue to (53) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℂ). capitalise on the rich legacy of Andrew Wiles’s marvelous The image of such a 휚 modulo the scalar matrices proof. is isomorphic either to a dihedral group, to 퐴4, to 푆 , or to 퐴 . Thanks to the earlier work of Hecke, 4 5 References Langlands, and Tunnell, only the case of projective [Da] H. Darmon, A proof of the full Shimura-Taniyama-Weil con- image 퐴 remained to be disposed of. Many new cases 5 jecture is announced, Notices of the AMS 46 (1999), no. 11, of the two-dimensional Artin conjecture were proved 1397–1401. MR1723249 in this setting by Kevin Buzzard, Mark Dickinson, Nick [Se] J-P. Serre, Lectures on 푁푋(푝), Chapman & Hall/CRC Re- Shepherd-Barron, and Richard Taylor around 2003 us- search Notes in Mathematics, 11, CRC Press, Boca Raton, ing the modularity of all mod 5 Galois representations FL. MR2920749 arising from elliptic curves as a starting point. • Serre’s conjecture, which was formulated in 1987, Photo Credits asserts the modularity of all odd, two-dimensional Photo of Wiles giving his first lecture is by C. J. Galois representations Mozzochi, courtesy of the Simons Foundation.

(54) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(픽푝푟 ), Photo of the conference in honor of Karl Rubin’s sixtieth birthday is courtesy of Kartik Prasanna. with coefficients in a finite field. This result was proved by Chandrasekhar Khare and Jean-Pierre Wintenberger in 2008 by a glorious extension of the “3-5 switching technique” in which essentially all the primes are used. (See Khare’s report in this volume.) This result also implies the two-dimensional Artin conjecture in the general case. • The two-dimensional Fontaine–Mazur conjecture con- cerning the modularity of odd, two-dimensional 푝-adic Galois representations

(55) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℚ̄ 푝) satisfying certain technical conditions with respect to their restrictions to the Galois group of ℚ푝. This theorem was proved in many cases as a consequence of work of Pierre Colmez, Matthew Emerton, and Mark Kisin.

216 Notices of the AMS Volume 64, Number 3 The Mathematical Works of Andrew Wiles Christopher Skinner, with contributions from Karl Rubin, Barry Mazur, Mirela Çiperiani, Chandrashekhar Khare, and Jack Thorne

ir Andrew J. Wiles was awarded the Abel Prize for 2016 for “his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in Snumber theory.”1 Andrew Wiles announced his proof of Fermat’s Last Theorem in June of 1993 in a series of three lectures at a conference at the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge. Overnight Wiles and his proof became an international media sensation, making headlines in papers around the world. The story of this proof—the subsequent discovery of a gap and its ultimate and beautiful completion in September of 1994—has entered into popular legend.2 The surprising drama of the proof is told in the 1996 BBC Horizon documentary Fermat’s Last Theorem, directed by Simon Singh, which ably conveys Andrew Wiles with his PhD advisor, John Coates (left); the human side of what is often seen as the distant and his first PhD student Karl Rubin (right); and Ken Ribet (second from right), whose work linking the rarefied world of mathematical research.3 Modularity Conjecture to Fermat’s Last Theorem All this is well known. What is less well known, possibly inspired Wiles to reengage with the problem that had even among number theorists, is that before his proof fascinated him since childhood. This photo was taken of Fermat’s Last Theorem, Wiles had made significant at the Newton Institute in Cambridge, UK, during the contributions to two of the most important problems for conference at which Wiles first announced his proof late-twentieth-century number theory: of the Modularity Conjecture.

Christopher Skinner is professor of mathematics at Princeton Uni- versity. His e-mail address is [email protected]. • Wiles proved, together with John Coates, the first the- 1Citation by the Abel Prize Committee of the Norwegian Acad- oretical evidence for the celebrated Birch–Swinnerton- emy of Science and Letters for the 2016 Abel Prize Laureate: www. Dyer Conjecture; this is now known as the Coates– abelprize.no/c67107/binfil/download.php?tid=67059 Wiles Theorem. • Wiles proved Iwasawa’s Main Conjecture for ℚ, in joint 2No doubt many number theorists share my own experiences of striking up conversations with strangers, who, upon discovering work with Barry Mazur, and for all totally real fields. that I am a mathematician and even a number theorist, ask about Each of these is a landmark result on its own and would “that guy who solved that famous problem—the one who worked be considered the highlight of a distinguished career. in his attic for seven years.” Two of the following contributions describe these works 3I have watched this documentary many times with groups of and their proofs. Karl Rubin, Wiles’s first PhD student, mathematically talented high school students from around the writes about the Coates–Wiles Theorem. Barry Mazur, world. Twenty years later it still inspires questions and conversa- Wiles’s collaborator on his first proof of Iwasawa’s Main tion about what it means to do mathematical research or what a Conjecture for ℚ, writes about Wiles’s work on the Main life spent doing mathematics can be. Conjectures. Anyone seeking to learn about the context, For permission to reprint this article, please contact: significance, and ideas of Wiles’s proof of Fermat’s Last [email protected]. Theorem can do no better than to read Henri Darmon’s DOI: http://dx.doi.org/10.1090/noti1485 Abel Prize lecture in this same issue of the Notices.

March 2017 Notices of the AMS 217 What are you working on now? This is a question that be seen as a vast generalization of the ideas in the proof eminent mathematicians are frequently asked (or so I am of the Coates–Wiles theorem. My own work with Eric reliably informed) and certainly one that Andrew Wiles Urban (2014), which together with Kato’s result proves has repeatedly faced in the years following his proof of much of the Main Conjecture in the Iwaswa theory of Fermat’s Last Theorem. In her contribution below, Mirela elliptic curves, is in large part the natural generalization Çiperiani writes about her collaboration with Andrew to some unitary groups of Wiles’s methods for proving the Wiles, from the mid-2000s, on a very natural Diophantine Iwasawa Main Conjecture for totally real fields. Andrew question: does a genus one curve over ℚ have a rational Wiles’s ideas continue to inform and shape progress on point over a solvable extension of ℚ? As Çiperiani explains, some of the fundamental problems in algebraic number results and techniques arising from the proof of Fermat’s theory. Last Theorem also play a role in this work. What distinguishes a great mathematical proof? There Karl Rubin is, of course, no definitive answer to this question. Cer- tainly proofs of famous or important open conjectures Wiles’s Work on Elliptic Curves with Complex can lay claim to being great. By this measure, Andrew Multiplication Wiles’s proof of Fermat’s Last Theorem is a truly great During my senior year at Princeton, 1975–76, there was proof. But proofs that introduce new ideas or open doors a lot of buzz in the common room about something to progress on problems that were previously viewed as everyone referred to as “Coates–Wiles.” I had a vague idea out of reach also have their claim to greatness. As noted in the Abel Prize citation, Andew Wiles’s proofs also achieve of what an elliptic curve was, but I doubt that I had any greatness by this second measure. clear idea of what John Coates and Andrew Wiles had The new techniques and ideas that led to a suc- done. But I could tell that it was important, and I had a cessful proof of the modularity of semistable elliptic mental image of two very senior mathematicians who had curves have been remarkably robust, also leading to the made a great breakthrough. resolution of a host of problems within the circle of I arrived at Harvard as a graduate student in 1976. the Langlands Program: proofs of Serre’s conjecture, the Wiles arrived a year later, and I discovered that this Artin conjecture for odd two-dimensional representations, “very senior” mathematician was scarcely older than I the Sato–Tate conjecture, meromorphic continuation of was and had been a graduate student at the time of Hasse–Weil 퐿-functions of elliptic curves over totally real this spectacular result. I became his student and had fields, modularity of all elliptic curves over real quadratic the unusual experience of attending my advisor’s thesis fields, and automorphy of small symmetric powers of defense. modular forms, to list just a few. Here’s what the excitement was about. An elliptic curve Much of this progress has been achieved by the efforts 퐸 over the field ℚ of rational numbers is a curve defined by an equation 푦2 = 푥3 + 푎푥 + 푏, with 푎, 푏 ∈ ℚ satisfying of Richard Taylor, who was a PhD student of Wiles and 3 2 later a coauthor of the paper that introduced one of the 4푎 + 27푏 ≠ 0. It is classical that the rational points 4 퐸(퐹) on 퐸 over any field 퐹 containing ℚ form an abelian key ingredients of the proof of modularity; by Wiles’s group, and Mordell proved that 퐸(ℚ) is finitely generated. PhD students; and by Taylor’s collaborators and PhD The rank of 퐸 is the dimension of the ℚ-vector space students. But Wiles’s work also inspired many others to 퐸(ℚ) ⊗ ℚ. push his ideas further. Among these is Chandrashekhar ℤ Khare, who writes below about some of the progress In the late 1950s, after extensive computations, Birch on modularity of two-dimensional Galois representations and Swinnerton-Dyer made the following conjecture. that followed Wiles’s proof and of the use of this in Conjecture (Birch & Swinnerton-Dyer). For every elliptic Khare’s own proof, with Jean-Pierre Wintenberger, of curve 퐸 over ℚ, we have Serre’s conjecture. In a related contribution, Jack Thorne (BSD) rank(퐸(ℚ)) = ord 퐿(퐸, 푠). describes how Wiles’s original techniques evolved and 푠=1 were adapted to proving the automorphy of higher- Here 퐿(퐸, 푠) is the Hasse–Weil 퐿-function of 퐸, defined dimensional Galois representations, leading to his proof, by an Euler product that converges on the complex with Laurent Clozel, of the automorphy of some small half-plane ℜ(푠) > 3/2, and ord푠=1 퐿(퐸, 푠) is its order of symmetric powers of a holomorphic modular form. vanishing at 푠 = 1. This conjecture is still unproved and But we should not lose sight of Wiles’s earlier contribu- is one of the Clay Millennium Problems. tions in the glow of the successes arising from the proof Clearly, to make progress on this conjecture, or even of Fermat’s Last Theorem. The proof of the Coates–Wiles for the statement to make sense, one needs to know that Theorem and Wiles’s proof of Iwasawa’s Main Conjec- 퐿(퐸, 푠) has an analytic continuation at least to 푠 = 1. This ture for totally real fields have inspired similar progress. was already known, thanks to a theorem of Deuring, for For example, Kazuya Kato’s (2004) spectacular success elliptic curves with complex multiplication: we say that 퐸 in constructing an Euler system for elliptic curves and has complex multiplication if the ring of endomorphisms then relating it to the special values of the Hasse–Weil End(퐸) is larger than ℤ, in which case End(퐸) ⊗ℤ ℚ is an 퐿-function of the curve via an explicit reciprocity law can Karl Rubin is Edward and Vivian Thorp Professor of Mathematics 4Now generally known as the Taylor–Wiles method. at UC Irvine. His e-mail address is [email protected].

218 Notices of the AMS Volume 64, Number 3 imaginary quadratic field. (The case End(퐸) = ℤ is much Conjecture and on the modularity of elliptic curves (see more common.) For example, the elliptic curve 푦2 = 푥3+푎푥 the contributions by Barry Mazur and Henri Darmon, has complex multiplication because (푥, 푦) ↦ (−푥, 푖푦) is respectively, in this issue), have continued to play an an automorphism of 퐸 of order 4. Deuring proved that if 퐸 important role in progress on the Birch and Swinnerton- has complex multiplication, then 퐿(퐸, 푠) can be identified Dyer conjecture. with the 퐿-function attached to a Hecke character and • Kolyvagin (1990) recognized that elliptic units form hence has an analytic continuation to the entire complex what he calls an Euler system. (In fact, as one of very plane. Further, in this case Damerell proved that there is few known examples, elliptic units helped him to an explicit Ω ∈ ℝ× such that 퐿(퐸, 1)/Ω ∈ ℤ. formulate the concept of an Euler system.) Combining By the mid-1970s little was known about (BSD) beyond the methods of Coates and Wiles with Kolyvagin’s computational examples. That was how things stood when Euler system machinery led to my 1991 proof of Coates and Wiles made their breakthrough: Iwasawa’s Main Conjecture for imaginary quadratic fields. Theorem 1 (Coates and Wiles, 1977 [2]). Suppose 퐸 has • Using a quite different Euler system of Heegner complex multiplication. If 퐸(ℚ) is infinite, then 퐿(퐸, 1) = 0. points, the combined results in the 1980s of Koly- In other words, if rank(퐸(ℚ)) > 0, then ord푠=1 퐿(퐸, 푠) > vagin, Gross–Zagier, Bump–Friedberg–Hoffstein, and 0. Murty–Murty showed that (BSD) holds if 퐸 is modular The key to the proof of Theorem 1 is the use of and ord푠=1 퐿(퐸, 푠) ≤ 1. Robert’s elliptic units to provide the crucial link between • The work of Wiles on modularity [6], completed by the algebraic and analytic sides of (BSD). Elliptic units are Taylor–Wiles [4] and Breuil–Conrad–Diamond–Taylor global units in abelian extensions of imaginary quadratic [1], showed in the 1990s that every elliptic curve over fields, defined by analytic functions, so they live in both ℚ is modular. Hence we have the following result for the algebraic and analytic worlds. all elliptic curves over ℚ, with or without complex Suppose 퐸 has complex multiplication, and let 퐾 multiplication. be the imaginary quadratic field End(퐸) ⊗ ℚ. Suppose ℤ Theorem 2. If ord 퐿(퐸, 푠) ≤ 1, then rank(퐸(ℚ)) = 푝 ≥ 5 is a prime where 퐸 has good reduction, and 푠=1 ord 퐿(퐸, 푠). 푝 factors as 푝 = 휋 ̄휋∈ End(퐸) ⊂ 퐾. For 푛 ≥ 1 let 푠=1 퐸[휋푛] ⊂ 퐸(퐾)̄ denote the kernel of the endomorphism Along with some Iwasawa-theoretic results due to Kato 푛 푛 휋 , let 퐾푛 ∶= 퐾(퐸[휋 ]), and let (2004) and Skinner and Urban (2014), this is currently the best result in the direction of the Birch and Swinnerton- Γ ∶= Gal(퐾 /퐾) ≅ (ℤ/푝푛ℤ)×. 푛 푛 Dyer conjecture for elliptic curves over ℚ. Since 퐾푛 is an abelian extension of 퐾, there is a subgroup × 풞푛 ⊂ 퐾푛 of elliptic units. We also let 푈푛 denote the units References in the ring of integers of the completion of 퐾푛 at the [1] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the unique prime above 휋. modularity of elliptic curves over ℚ: wild 3-adic exercises, Coates and Wiles constructed a “logarithmic derivative” J. Amer. Math. Soc. 14 (2001), 843–939. MR1839918 푛 homomorphism 휓푛 ∶ 푈푛 → 퐸[휋 ] and computed that [2] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 223–251. 퐿(퐸, 1) 푛 휓푛(풞푛) = 퐸[휋 ]. MR0463176 Ω [3] K. Rubin, The “main conjectures” of Iwasawa theory for Under a mild additional assumption on 푝, they showed imaginary quadratic fields, Invent. Math. 103 (1991), 25–68. that the group of Γ푛-equivariant homomorphisms MR1079839 푛 푛 [4] C. Skinner and E. Urban, The Iwasawa main conjectures for HomΓ푛 (푈푛, 퐸[휋 ]) is cyclic of order 푝 , generated by 휓푛, and therefore GL2, Invent. Math. 195 (2014), 1–277. MR3148103 [5] R. Taylor and A. Wiles, Ring-theoretic properties of cer- 푛 푛 퐿(퐸,1) (1) HomΓ푛 (푈푛/풞푛, 퐸[휋 ]) ≅ ℤ/(푝 , Ω )ℤ. tain Hecke algebras, Ann. of Math. 141 (1995), 553–572. Now suppose 퐸(ℚ) is infinite. Fix a point 푃 ∈ 퐸(ℚ) of MR1333036 [6] A. Wiles, Modular elliptic curves and Fermat’s last theorem, infinite order, and for every positive integer 푛 choose a 푛 Ann. of Math. 141 (1995), 443–551. MR1333035 point 푄푛 ∈ 퐸(퐾)̄ such that 휋 (푄푛) = 푃. The “Kummer map” that sends 휎 ∈ Gal(퐾/퐾̄ 푛) to 휎(푄푛) − 푄푛 defines a homomorphism Barry Mazur 휅 ∈ Hom (Gal(퐾/퐾̄ ), 퐸[휋푛]). 푛 Γ푛 푛 Andrew Wiles’s Work on the Main Conjecture Using class field theory, 휅푛 induces a homomorphism of Iwasawa Theory 푛 What a joy it was to work with Andrew on the Main 휅푛̃ ∈ HomΓ푛 (푈푛/풞푛, 퐸[휋 ]), Conjecture of Iwasawa theory over ℚ, and how great that and Coates and Wiles showed that there is an integer 푘, 푛−푘 Andrew went on to establish it, more generally, for totally independent of 푛, such that 휅푛̃ has order 푝 for all 푛 ≥ 푘. real fields. Comparing this with (1) as 푛 grows proves Theorem 1. The ideas in the proof of Theorem 1, along with Barry Mazur is Gerhard Gade University Professor at Harvard methods Wiles developed for his work on Iwasawa’s Main University. His e-mail address is [email protected].

March 2017 Notices of the AMS 219 푎 The basic idea behind the Main Conjecture might be 휁 ↦ 휁 for any 휁 ∈ 휇푛. Any automorphism of 휇푛 extends thought of as having, as a starting place, the classical uniquely to an automorphism of the field ℚ[휇푛], giving analytic formulas of number theory, such as Dirichlet’s us canonical isomorphisms: Class Number Theorem for quadratic imaginary fields: ∗ ≃ ≃ (ℤ/푛ℤ) ⟶ 퐴푢푡(휇푛) ⟶ Gal(ℚ[휇푛]/ℚ). 퐿(휒, 1) ℎ(−푑) = , Fixing a prime 푝 (for simplicity, suppose 푝 > 2) and letting 2휋 푤√푑 푛 run through powers of 푝, form the 푝-cyclotomic tower where 휒 is the quadratic Dirichlet character cutting out the of (abelian Galois) extensions quadratic imaginary field ℚ(√−푑) of (integer) conductor ℚ ⊂ ℚ[휇 ] ⊂ ℚ[휇 2 ] ⊂ ℚ[휇 3 ] ⊂ … −푑 < 0; ℎ(−푑) is the order of the ideal class group of that 푝 푝 푝 ∞ field; 푤 is the number of roots of unity in it; and 퐿(휒, 푠) and put ℚ[휇푝∞ ] ∶= ⋃휈=1 ℚ[휇푝휈 ]. We have that 휈 ∗ ∗ is the Dirichlet 퐿-function. One of the striking aspects Gal(ℚ[휇푝∞ ]/ℚ) = lim (ℤ/푝 ℤ) = ℤ푝 , of this formula is that the left-hand side is analytic; the 휈→∞ ∗ right-hand side is arithmetic. the latter, ℤ푝 , being the profinite topological group of ∗ The “Main Conjecture” is not a misnomer: for any 푝-adic units, which decomposes as a product, ℤ푝 = ∗ ∗ prime number 푝 the conjecture asserts a fundamental F푝 ×{1+푝ℤ푝}, where F푝 ≃ Gal(ℚ[휇푝]/ℚ) is a cyclic group ∗ relationship that establishes a close tie between of order 푝−1, and the subgroup Γ ∶= {1+푝Z푝} ⊂ ℤ푝 is an • the 푝-adic 퐿-functions of a number field 푘—these being infinite cyclic pro-푝-group. A neat topological generator ∗ 푝-adic analytic objects closely related to the classical to choose for Γ is the 푝-adic unit 훾 ∶= (1 + 푝) ∈ Γ ⊂ ℤ푝 . (complex) 퐿-functions of 푘, The field ℚ[휇푝∞ ] is generated by two linearly disjoint • the deep arithmetic of that number field—namely, the subfields: ℚ[휇푝] and a field, call it ℚ∞, Galois over ℚ with 푝-primary parts of ideal class groups of certain abelian Galois group ∗ extensions of 푘. Gal(ℚ∞/ℚ) = Γ ∶= 1 + 푝Z푝 ⊂ Z푝 . Slightly more specifically, the Main Conjecture identifies The topological group Γ is our “group of operators.” the zeroes of 푝-adic 퐿-functions of a number field 푘 with the eigenvalues of an operator on a 푝-adic vector The Vector Space Containing Basic Arithmetic space constructed from the 푝-primary parts of ideal Data Related to Number Fields class groups of the abelian extensions of 푘 alluded to above or constructed from some closely related arithmetic Let 푘 ⊂ 퐾 be number fields, contained in 풞, linearly disjoint from ℚ , with 퐾 totally real, and 퐾/푘 a cyclic objects.1 ∞ Galois extension. Put 퐾 = 퐾 ⋅ ℚ ⊂ 풞. So Gal(퐾 /퐾) = This puts the conjecture somewhat in the spirit of a ∞ ∞ ∞ Gal(ℚ /ℚ) = Γ. suggestion attributed to Hilbert and Pólya that a complex ∞ Let 퐿 denote the maximal unramified pro-푝 abelian 퐿-function of a number field might arise naturally as ∞ extension of 퐾 . Let 푋 ∶= Gal(퐿 /퐾 ), which, since it is a the characteristic series attached to a certain unbounded ∞ ∞ ∞ projective limit of 푝-abelian groups, we can view naturally operator on a (naturally defined) Hilbert space. (Their as a ℤ -module. Also, Γ = Gal(퐾 /퐾) acts naturally (and suggestion, though, goes further by noting that if these 푝 ∞ ℤ -linearly) on 푋. The action of an element in Γ on operators were self-adjoint, this would relate well to 푝 푋 is defined by lifting it to an element in Gal(퐿 /퐾) the Riemann Hypothesis.) It also is in the spirit of the ∞ and then noting that conjugation by that lifted element classical theory of 퐿-functions attached to varieties over doesn’t depend on the lifting and induces a well-defined finite fields, for such an 퐿-function can also be thought of automorphism of 푋. as the characteristic polynomial of the Frobenius operator ̄ One knows that 푉 ∶= 푋 ⊗ℤ푝 ℚ푝 is a finite-dimensional on an appropriate étale cohomology group. ∗ To outline an example of the Main Conjecture, we’ll vector space. If 휒 ∶ Gal(퐾/푘) ↪ 풞 is an odd (faithful) discuss the relevant group of operators, vector space, 푝- character cutting out the field extension 퐾/푘, we will be adic 퐿-functions, and the method for the construction of considering the action of Γ on the 휒-part of 푉, i.e., on 푉휒 ∶= {푣 ∈ 푉 | 푔(푣) = 휒(푔) ⋅ 푣}. We want to get as full an the relevant arithmetic objects. understanding of the 푉휒 as possible and specifically the eigenvalues of 1 − 훾 on 푉휒 where 훾 ∈ Γ is a topological The 푝-Cyclotomic Tower and the Fundamental generator. Operator in the Main Conjecture Let 푛 ≥ 1. Consider the finite field extension ℚ[휇푛]/ℚ 퐿-Functions obtained by adjoining the group of 푛th roots of unity, By interpolating special values of the classical complex 2휋푖푎/푛 ∗ 휇푛 ∶= {푒 | 푎 = 0, 1, … , 푛 − 1} ⊂ 풞 , 퐿-functions, Kubota and Leopoldt defined the 푝-adic 퐿- to the field ℚ of rational numbers. The group (ℤ/푛ℤ)∗ is functions over the field 푘 = ℚ, and subsequently Deligne and Ribet defined them over totally real fields 푘. Briefly, canonically isomorphic to the group 퐴푢푡(휇푛) of automor- in the context above, let 휁 (휎, 푠) denote the partial phisms of the cyclic group 휇푛, this isomorphism being 푘 defined by sending 푎 ∈ (ℤ/푛ℤ)∗ to the automorphism zeta-function of 푘 associated to elements 휎 ∈ Gal(퐾/푘). The special values 휁푘(휎, 1 − 푛) are rational numbers, as 1For example, from abelian extensions of 푝-power degree unrami- proved by Klingen and Siegel. Let 휓 be a one-dimensional ∗ −1 푛 fied over those alluded-to extensions of 푘. character over 푘 with values in ℚ푝 and 휒푛 ∶= 휓 휔

220 Notices of the AMS Volume 64, Number 3 where 휔 is the Teichmüller character. For any integer References 푛 ≥ 1 one puts [1] B. Mazur and A. Wiles, Class fields of abelian extensions of ℚ, Invent. Math. 76 (1984), 179–330. MR742853 퐿푝(1 − 푛, 휓) [2] B. Mazur, How can we construct abelian extensions of number fields?, Bulletin of the A.M.S. 48 (2011), 155–209. = 휒 (휎)휁 (휎, 1 − 푛) ⋅ (1 − 휒 (푃)푁(푃)1−푛), ∑ 푛 푘 ∏ 푛 MR2774089 휎∈Gal(퐾/푘) 푃 |푝 [3] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493–540. MR1053488 where “∏푃 |푝” is taken over all primes 푃 of 푘 lying above 푝, and 푁(푃) is the norm of the ideal 푃. These special values, {1 − 푛 ↦ 퐿푝(1 − 푛, 휓) ∈ ℚ̄ 푝}, Mirela Çiperiani interpolate to produce a 푝-adic analytic function 퐿푝(푠, 휓) on ℤ푝 if 휓 is not trivial and on ℤ푝 − {1} with a simple Solvable Points on Genus One Curves pole at 푠 = 1 when 휓 is trivial. Moreover, there is a In the spring of 2002, as a graduate student at Princeton, unique power series 퐺휓(푇) ∈ ℤ̄푝[[푇]] such that for the I approached Andrew Wiles to ask about the possibility ∗ topological generator 훾 ∈ Γ ⊂ ℤ푝 (viewed as an element of working under his supervision. He suggested that ∗ of ℤ푝 ) we have I think about a problem that was thrillingly natural and beautiful: Does every genus one curve, defined over 푠 퐿푝(푠, 휓) = 퐺휓(훾 − 1). the rational numbers ℚ, have a point over some solvable extension of ℚ? The Main Conjecture, then, for 휒 odd and 푝 > 2 Solvable extensions of ℚ—Galois extensions with solv- identifies the characteristic polynomial of 훾 acting on 푉휒 able Galois group—are, concretely, fields contained in as described above, with the Weierstrass polynomial of −1 −1 root extensions of ℚ, i.e., in fields obtained by starting the power series 퐺휓(훾(1 + 푇) − 1) where 휓 = 휒 휔. with ℚ and considering successive extensions of the That is, as Andrew proves [3], the eigenvalues of 훾 acting form F( 푛 훼)/F for some 훼 ∈ F and 푛 ∈ ℕ. This relation- 휒 √ on 푉 are identified with the zeroes of 퐿푝(푠, 휓). ship between solvability of the Galois group and iterated adjunction of roots is the Galois-theoretic criterion for Method solvability by radicals of a polynomial equation in one The essential issue in proving the Main Conjecture is to variable. Thus, as shown by Abel and Galois, equations construct (by means of Galois representations attached to in one variable with coefficients in ℚ and degree at modular forms) as many abelian unramified extensions least 5 need not be solvable by radicals. This connection as would be predicted from the analytic side of the to classical Galois theory is one of the appeals of the conjectured formula. A version of the classical analytic problem. formula then allows one to conclude the conjecture. When An obvious stumbling block is that, while this is evi- the field 푘 is ℚ, we did this [1] by a two-step approach, dently a diophantine equations problem, it is much less tangible than that of solving a quintic: we don’t have a starting from a marvelous idea of Ken Ribet relating generic way of writing the equations that describe genus divisibility of the 푝-adic 퐿-function by 푝 to a similar one curves defined over ℚ. A priori, a genus one curve divisibility of the order of a specific ideal class group defined over ℚ is cut out by several homogeneous poly- by 푝. (For a leisurely discussion of Ribet’s method and nomial equations with rational coefficients. The famous connection with the earlier work of Herbrand, see [2].) Weierstrass cubic equations 푦2푧 = 푥3 + 푎푥푧2 + 푏푧3 (here Briefly, the 푝-adic 퐿-function 퐿푝(푠, 휒) (times an elementary 푎, 푏 ∈ ℚ such that 푥3 + 푎푥 + 푏 has no repeated roots) de- nonzero factor) occurs as the constant term of a 푝-adic scribe elliptic curves, i.e., genus one curves which do have Eisenstein series (of 푝-adic weight determined by the a point over their field of definition, namely, (0 ∶ 1 ∶ 0). value of 푠). Whenever the 푝-adic 퐿-function vanishes, However, each genus one curve is a torsor for an elliptic this Eisenstein series has constant term zero and can be curve, its Jacobian; i.e., the Jacobian acts on the genus one shown, therefore, to be congruent modulo arbitrarily high curve, and over ℚ that action becomes freely transitive. powers of 푝 to cuspidal eigenforms. Moreover, since these Conversely, every torsor for an elliptic curve E is a genus cuspidal modular eigenforms are congruent modulo a one curve with Jacobian E. In light of this correspon- high power of 푝 to Eisenstein series, their associated 푝-adic dence, genus one curves over ℚ with fixed Jacobian E are Galois representations are extremely well behaved modulo parametrized by a (Galois) cohomology group, the Weil– 1 those powers of 푝 and can be seen to cut out larger and Châtelet group H (Gal(ℚ/ℚ),̄ E(ℚ))̄ . So now we dispense with defining equations of genus one curves and work larger unramified 푝-power abelian extensions of 퐾∞. The procedure employed by Wiles [3] for the totally real case is with elements of the Weil–Châtelet group. a good deal more delicate than in the case of 푘 = ℚ, in that In 2006, in joint work with Wiles, we showed that one now uses the Galois representations of eigenforms under certain restrictions, Wiles’s original question has a positive answer, as stated momentarily. The removal of on Hilbert–Blumenthal moduli spaces to construct the these restrictions is the subject of ongoing joint work. desired unramified abelian extensions. Andrew works systematically over the relevant weight space constructing Mirela Çiperiani is associate professor of mathematics at the the appropriate cuspidal Λ-adic Hilbert modular forms, University of Texas at Austin. Her e-mail address is mirela@ using Hida’s theory, and [3] proves much more. math.utexas.edu.

March 2017 Notices of the AMS 221 Theorem ([ÇW]). Let C be a genus one curve defined over sufficiently many cohomology classes ramified at ℚ, with Jacobian E, such that primes in 풬 depends on the finiteness of all the

(1) C has a point defined over the ℓ-adics ℚℓ for all 푝-primary components of the Tate–Shafarevich rational primes ℓ; and group of E/ℚ. This is only known for elliptic (2) one of the following conditions holds: curves E/ℚ of analytic rank less than or equal to (a) the analytic rank of E/ℚ is less than or equal 1 (by the modularity theorem, and the combined to 1 or and celebrated work of Kolyvagin, Gross–Zagier, (b) E is semistable over ℚ. Bump–Friedberg–Hoffstein, and Murty–Murty). Then C has a point over a solvable extension of ℚ. When conditions (1) and (2b) hold, we can still find nontrivial Heegner points, using work of Cornut-Vatsal, The analytic rank of an elliptic curve E is the order by viewing C as a genus one curve over a nontrivial of vanishing of its 퐿-series 퐿(E, 푠) at 푠 = 1. As for extension of ℚ. We now attempt to apply the unramified semistability, after lifting an elliptic curve from ℚ to ℤ and under ramified principle. However, while this field exten- then reducing modulo a prime, it may remain nonsingular sion enables us to construct ramified Kolyvagin classes, or it may acquire a node or cusp. Semistability disallows because of issue (ii) we are not able to see that we have cusps. sufficiently many of them. I will now describe two fundamental ideas of Wiles It is in circumventing the potential existence of an which provide the frame of the proof. The first is referred infinite 푝-primary component of the Tate–Shafarevich to as the “unramified under ramified principle.” Fix an group that we need the second idea. It is referred to as elliptic curve E defined over ℚ. Genus one curves C defined the “patching method.” A similar method was used in over ℚ with E as their Jacobian, and with points over the proof of the modularity theorem. Wiles’s idea is the ℚℓ for every rational prime ℓ, form a subgroup of the following. We construct as many ramified classes as we Weil–Châtelet group. (This subgroup contains the Tate– can for each in an infinite sequence of field extensions Shafarevich group, which consists of genus one curves {퐹푛}. Observe that in order to do this we must choose that have both ℓ-adic and real points.) We in fact work with unrelated sets of primes 풬푛 for the fields 퐹푛, all with the their preimages in the Selmer group. These Selmer classes same cardinality. We consider groups 푀푛 of cohomology become trivial after an extension of ℚ unramified at all classes over 퐹푛 ramified at primes in 풬푛 (actually, 푀푛 but a finite set of primes ℓ. According to the unramified is viewed as a module over a ring related to 퐹푛). There under ramified principle, if we find a finite set of primes is no natural containment between these modules, but 풬 obeying certain conditions and if we have sufficiently each of them contains all the classes over ℚ that we want many cohomology classes which are ramified at primes to capture. However, our Kolyvagin classes generate a ′ ′ in 풬, then under the cohomology group structure they submodule 푀푛 ⊆ 푀푛, and for no 푛 can we see that 푀푛 will generate all the unramified classes. contains the desired classes. By considering their mod- We will apply this principle in the case where the genus ule structure (and ignoring their content), we construct one curve C satisfies conditions (1) and (2a) of our theorem injective maps 푀푛 → 푀푛+1. This gives rise to a module by constructing ramified classes which correspond to lim 푀 over an Iwasawa algebra. Miraculously, a structure ⎯⎯⎯⎯⎯→ 푛 genus one curves with points over some solvable extension theorem from Iwasawa of ℚ. These ramified classes will then generate a group theory shows us that containing all genus one curves that have the same I developed each of our genus Jacobian as C and satisfy conditions (1) and (2a). Hence one curves C satisfy- C has a solvable point. Thus the task is to choose the enormous ing conditions (1) and set 풬 and to construct sufficiently many ramified classes ′ (2b) lies in some 푀푛 for which have solvable points. This is achieved by using admiration of some 푛. Thus 퐶 has a the cohomology classes constructed by Kolyvagin. The solvable point. construction of these classes makes use of Heegner Wiles for his Working with Wiles points. These points are defined on modular curves and has been a wonderful pushed forward to the elliptic curve via its modular generosity and experience. At the be- parametrization—whose existence is known by the work modesty, and awe ginning, when I barely of Wiles extended by Breuil, Conrad, Diamond, and Taylor. knew what an ellip- The unramified under ramified principle is sufficient for his ability to tic curve was, I felt to prove the theorem in the case when the analytic rank lucky but humbled and of E/ℚ is less than or equal to 1. It is not sufficient to see to the heart of daunted to be en- prove it in general, for the following two reasons: trusted with such a (i) If the analytic rank of E/ℚ is greater than 1, then the matter. fantastic problem. I the relevant Heegner points are trivial. Hence, in had to rapidly learn this case we cannot construct enough nontrivial the background material needed to begin thinking about Kolyvagin classes. the problem and to understand Wiles’s proposal to use (ii) The set of primes 풬 depends on the order of the unramified under ramified principle. Later, at times the curve C viewed as an element of the Weil– when the problem seemed impossible, I was buoyed by Châtelet group of E/ℚ, and the existence of the Wiles’s confidence that we could solve it. I developed

222 Notices of the AMS Volume 64, Number 3 enormous admiration of Wiles for his generosity and ℓ. Wiles’s theorem “lifts modularity” in that it asserts that modesty, and awe for his ability to see to the heart of the if 휌̄ is modular, then the lift 휌 of 휌̄ is also modular. matter—feelings that stay with me today. Wiles’s theorem allows for representations 휌 where ℤ푝 is replaced by more general 푝-adic rings. In particular, References it allows for homomorphisms 휌 ∶ 퐺ℚ → GL2(ℚ푝); a [ÇW] M. Çiperiani and A. Wiles, Solvable points on genus one continuity argument associates to such a 휌 a residual curves, Duke Math. J. 142 (2008), 381–464. MR2412044 representation 휌̄ ∶ 퐺ℚ → GL2(픽푝). Much earlier results of Shimura and Deligne attach to each eigenform 푓 and Chandrashekhar Khare embedding 휄 ∶ ℚ → ℚ푝 a Galois representation

휌푓,휄 ∶ 퐺ℚ → GL2(ℚ푝). Modularity of GL2 Galois Representations The spirit of Wiles’s modularity lifting theorem is: if and the Work of Andrew Wiles 휌̄ ≅ 휌푓,휄̄ for some eigenform 푓, then 휌 ≅ 휌푔,휄′ for some The work of Andrew Wiles on the modularity of elliptic eigenform 푔 (not necessarily the same as 푓). More about curves provided some of the key ideas and techniques this theorem and its context can be found in Darmon’s which led to the proof of Serre’s modularity conjecture. lecture in this issue and in the article by Thorne. Moreover, and equally importantly, it psychologically Wiles’s strategy for proving his marvelous modularity made it possible to imagine that there could be a strat- lifting theorem can very roughly be paraphrased as follows. egy to prove results that before Wiles’s work seemed The kernel of the mod 푝 reduction map GL2(ℤ푝) → GL2(픽푝) completely inaccessible. is prosolvable (the inverse limit of solvable groups), which To retrace the path from Wiles’s work [5] on modularity enables the use of duality theorems in Galois cohomology of elliptic curves to the proof of Serre’s conjecture [2], we and congruences between modular forms to bootstrap describe briefly Wiles’s modularity lifting theorem and the modularity property from 휌̄ to 휌. his strategy for proving it. One spectacular application of Wiles’s modularity lift- Let 퐺ℚ = Gal(ℚ/ℚ) be the Galois group of an algebraic ing theorem was the proof of the modularity of semistable closure ℚ of ℚ. Wiles’s modularity lifting theorem is elliptic curves defined over ℚ (and hence Fermat’s Last about 2-dimensional 푝-adic Galois representations for a Theorem!). Elliptic curves are often encountered as the prime 푝 > 2. These are certain homomorphisms projective curves defined by an (affine) Weierstrass equa- 2 3 휌 ∶ 퐺ℚ → GL2(ℤ푝), tion: 푦 = 푥 + 푎푥 + 푏 for constants 푎 and 푏. The curve is defined over ℚ if 푎, 푏 ∈ ℚ. The points on an elliptic curve including those that arise from the action of 퐺ℚ on the torsion points of an elliptic curve defined over ℚ. Wiles 퐸 form an abelian group, with the addition law defined by proved [5], [4] that under certain hypotheses on 휌 and on rational functions in 푥 and 푦; the identity element of the its residual representation group is the unique point at infinity (the point (0 ∶ 1 ∶ 0) in projective coordinates). The torsion points 퐸[푁] on 퐸 휌̄ ∶ 퐺ℚ → GL2(픽푝) of (positive integer) order 푁 form a group isomorphic to (the reduction of 휌 modulo 푝), the representation 휌 is ℤ/푁ℤ × ℤ/푁ℤ. If the elliptic curve is defined over ℚ, then modular. the Galois group 퐺ℚ acts on 퐸[푁] by its action on the The hypotheses on the residual representation 휌̄ in coordinates of the points. The 푝-adic Tate module 푇푝퐸 of Wiles’s theorem require that 휌̄ be odd (the image of com- 퐸 is then the inverse limit of the groups 퐸[푝푛] of 푝-power 푛 2 plex conjugation has eigenvalues +1 and −1), irreducible, torsion points: 푇푝퐸 = lim 퐸[푝 ] ≅ ℤ푝. The action of 퐺ℚ ←⎯⎯⎯⎯⎯푛 and modular. For 휌,̄ being modular means that there is a on 푇푝퐸 determines a 푝-adic Galois representation cuspidal modular eigenform (an analytic function on the 휌 ∶ 퐺 → GL (ℤ ). complex upper-half-plane with many symmetries) 퐸,푝 ℚ 2 푝 ∞ The residual representation 휌퐸,푝̄ of 휌퐸,푝 is just the rep- 2휋푛휏 푓(휏) = ∑ 푎푛(푓)푒 resentation of 퐺ℚ on the group of 푝-torsion points 푛=1 2 퐸[푝] ≅ 픽푝. The modularity of an elliptic curve can be such that for all but finitely many primes ℓ, the Fourier interpreted as the 푝-adic Galois representation 휌퐸,푝 being coefficient 푎ℓ(푓) can be matched with the trace of 휌̄ modular for some prime 푝 (equivalently, all primes 푝). evaluated on a Frobenius element in 퐺ℚ for the prime ℓ. The application of the modularity lifting theorem to More precisely, the Fourier coefficients 푎푛(푓) are all alge- modularity of elliptic curves over ℚ comes by taking 휌 = braic integers (so belong to ℚ) and for some embedding 휌퐸,푝. For 푝 = 2 or 3 the image of the representation 휌퐸,푝 휄 ∶ ℚ ↪ ℚ푝 of ℚ into an algebraic closure ℚ푝 of ℚ푝, for is prosolvable (the inverse limit of finite solvable groups). almost all primes ℓ the image of 휄(푎ℓ(푓)) in the residue This is one of the two places in his argument where Wiles field 픽푝 of ℚ푝 equals the trace of 휌̄ on a Frobenius element uses lucky accidents which happen for small primes. He for ℓ. Similarly, 휌 is modular if there is an eigenform 푓 uses results of Langlands and Tunnell which imply that and an embedding 휄 such that for almost all primes ℓ, an odd irreducible representation 휌̄ ∶ 퐺ℚ → GL2(픽3) is modular. This uses the solvability of GL (픽 ) for 푝 = 3 휄(푎ℓ(푓)) equals the trace of 휌 on a Frobenius element for 2 푝 (indeed, PGL2(픽3) is isomorphic to 푆4) and the fact that Chandrashekhar Khare is professor of mathematics at UCLA. His the map GL2(ℤ푝) to GL2(픽푝) splits for 푝 = 3; both these e-mail address is [email protected]. statements are false for 푝 > 3!

March 2017 Notices of the AMS 223 At this point, the modularity lifting theorem allows these descent results imply that modularity of 휌̄ follows

Wiles to conclude modularity of semistable 퐸 unless by showing that 휌|̄ 퐺퐹 arises from a Hilbert modular form 휌퐸,3̄ is reducible. To deal with the case when 휌퐸,3 is for a solvable, totally real extension 퐹/ℚ. reducible, Wiles plays a “3-5” trick. He constructs another To study general representations 휌̄ as in Serre’s con- semistable elliptic curve 퐸′ over ℚ, whose mod 3 repre- jecture, Taylor considered moduli spaces over ℚ whose sentation surjects onto GL2(픽3) and such that the mod points over a number field 퐾 correspond to abelian va- 5 representations arising from 퐸 and 퐸′ are isomorphic rieties over 퐾 (with real multiplication) which give rise and irreducible. Here again the use of the small prime to 휌|̄ 퐺퐾 , and at an auxiliary place ℓ ≠ 푝 give rise to a 5 is vital, as the moduli space he considers of elliptic mod ℓ dihedral representation. The latter are known to curves with level 5 structure isomorphic to 퐸[5] turns be modular by an old result of Hecke. Taylor used a out to be of genus zero, and in fact the projective line theorem of Moret–Bailly to produce points of the moduli over ℚ, and thus has many rational points. Then applying spaces he considered over totally real fields 퐹 (but which the modularity lifting theorem he deduces that 휌퐸′,3 is could not be guaranteed to be solvable over ℚ). Then modular. This implies that 휌퐸′,5 is also modular, hence modularity lifting theorems for representations of 퐺퐹

휌퐸,5̄ = 휌퐸̄ ′,5 is modular, and then by another application yield that 휌|̄ 퐺퐹 arises from a Hilbert modular form over of the theorem, that 휌퐸,5, and therefore 퐸, is modular. 퐹. This may be regarded as a potential version of Serre’s Wiles’s work was generalized by Breuil, Conrad, Dia- conjecture. Together with automorphic descent for cyclic mond, and Taylor (2001) to prove the modularity of all prime degree extensions of totally real number fields, this elliptic curves over ℚ. Recently, Freitas, Hung, and Siksek led to the meromorphic continuation of the Hasse–Weil (2015) proved the modularity of elliptic curves defined 퐿-series attached to elliptic curves over all totally real over all real quadratic fields, using a “3-5-7” trick. fields. Before the work of Wiles there was no path from a 푝-adic To proceed along these lines to prove Serre’s conjecture Galois representation to a modular form. His completely in the general case, it would be necessary either to new method of modularity lifting showed that if only a find a general procedure to show existence of totally small quotient of a 푝-adic representation arose from a real solvable points on geometrically irreducible smooth modular form, then the entire Galois representation did. projective varieties of general type over ℚ or to prove This has turned out to be a very powerful method to nonsolvable automorphic descent for Hilbert modular prove modularity of Galois representations! forms.

Conjectures of Serre and Fontaine–Mazur Proof of Serre’s Conjecture My proof with Wintenberger of Serre’s conjecture [2] took Serre conjectured [3] that for any 휌̄ ∶ 퐺ℚ → GL2(픽푝) that is continuous, odd, and irreducible, there is an eigenform a different path and used as a starting point results of Tate and Serre that proved Serre’s conjecture for 푓 such that 휌̄ ≃ 휌푓,휄̄ ; i.e., 휌̄ is modular. J.-M. Fontaine and B. Mazur conjectured [1] that if 휌 ∶ representations 휌̄ of residue characteristic 푝 ≤ 3 with limited ramification, with no a priori assumptions on the 퐺ℚ → GL2(ℚ푝) is continuous, odd, irreducible, unramified outside a finite set of primes, and potentially semistable image of 휌.̄ Tate and Serre proved that any representation with Hodge–Tate weights (푎, 푏), say 푎 ≤ 푏, then the 휌̄ ∶ 퐺ℚ → GL2(픽푝) unramified outside 푝 and with 푝 ≤ 3 is cyclotomic twist 휌(−푎) is modular. reducible. This is another instance of the magic of small Wiles’s modularity lifting results were in the direction primes! of showing that the conjecture of Serre implies that of Our proof measured the complexity of the represen- Fontaine–Mazur. These modularity lifting results were tation 휌̄ in terms of its ramification at 푝 (Serre’s weight) improved in various crucial ways by Diamond, Fujiwara, and the ramification away from 푝 (its Artin conductor 푁). and Kisin, including generalizations with the base field A double induction on (푝, 푁) was used to reduce Serre’s ℚ replaced by a totally real field 퐹. In an important conjecture to the results of Tate and Serre. A principle of the proof is to use potential modularity to produce development, Skinner and Wiles lifted the condition that compatible systems of representations that lift 휌̄ such as 휌̄ is irreducible in the case when the lift 휌 is ordinary at would exist if it were known that 휌̄ were modular. 푝. All these developments were crucial in our later work Our proof of Serre’s conjecture owes its existence to on Serre’s conjecture. the modularity lifting theorems initiated by Wiles, a tool to attack modularity which was as powerful as it was Potential Version of Serre’s Conjecture unexpected when it was introduced, and also to Wiles’s The automorphic descent results of Saito and Shintani, and prime switching trick in his proof of the modularity of Langlands are an important ingredient in the applications elliptic curves. of modularity lifting theorems. These show that given a cyclic extension of totally real number fields 퐾/퐹 of References prime degree and a cuspidal automorphic representation [1] Jean-Marc Fontaine and Barry Mazur, Geometric Ga- of 휋 of GL2(픸퐾) which is a discrete series at the infinite lois representations, Elliptic Curves, Modular Forms, & places and invariant under 휎 ∈ Gal(퐾/퐹), then 휋 is the Fermat’s Last Theorem (Hong Kong, 1993), Ser. Number base change of a cuspidal automorphic representation of Theory, I, Internat. Press, Cambridge, MA, 1995, pp. 41–78. GL2(픸퐹). When combined with modularity lifting results, MR1363495

224 Notices of the AMS Volume 64, Number 3 [2] Chandrashekhar Khare and Jean-Pierre Wintenberger, Passing to spectra, one can think of Spec 핋휚,푆 as a closed Serre’s modularity conjecture (I), Invent. Math. 178 (2009), subspace of Spec 푅휚,푆: it is the locus of the modular no. 3, 485–504. MR2551763 Galois representations inside the space of all Galois [3] Jean-Pierre Serre, Sur les représentations modulaires de de- representations. In particular, the existence of this map gré 2 de Gal(Q/Q), Duke Math. J. 54 (1987), no. 1, 179–230. expresses the existence of Galois representations attached MR885783 to modular forms, itself a highly nontrivial fact. Diagrams [4] Richard Taylor and Andrew Wiles, Ring-theoretic proper- ties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), such as (4) have achieved an iconic status in algebraic no. 3, 553–572. MR1333036 number theory. [5] Andrew Wiles, Modular elliptic curves and Fermat’s last The truth of the modularity lifting theorem is implied theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. by the much more refined statement that the map (4) is an MR1333035 isomorphism. This goes some way towards explaining the importance of the universal deformation ring 푅휚,푆, which Jack Thorne was first introduced by Mazur. A large part of Wiles’s fundamental 1995 work is taken up with introducing the Modularity of 푛-Dimensional Galois Repre- tools necessary to effectively study the map (4), putting sentations him in a position to prove ‘푅 = 핋’ in many cases. One tool that has turned out to be surprisingly versatile More than twenty years have passed since Andrew Wiles is the Taylor–Wiles method, introduced in the companion proved the first modularity lifting theorems and deduced paper [3]. Roughly speaking, this is an effective machine to Fermat’s Last Theorem as a consequence. His theorems fo- study (4) in the so-called minimal case where 푆 is as small cussed on proving the modularity of certain 2-dimensional as possible. To pass from this case to the general case, Galois representations. These were the homomorphisms Wiles introduced a numerical isomorphism criterion to (1) 휚 ∶ 퐺ℚ,푆 → GL2(ℤℓ) compare the situation for varying sets 푆. The verification arising from elliptic curves 퐸, 퐺ℚ,푆 being the Galois group of this criterion then involves delicate calculations in of the maximal extension of ℚ unramified outside some Galois cohomology and with modular forms. finite set of primes 푆. Despite this focus, the ideas of the The first modularity lifting theorems for Galois rep- proof have proved powerful and flexible enough that they resentations of dimension 푛 > 2 were proved by Clozel, are still a driving force of our understanding of much Harris, and Taylor in a paper published in 2008, which was more general Galois representations today. heavily influenced by an earlier unpublished manuscript The first key hypothesis imposed on 휚 is the modularity of Harris and Taylor. This was made possible thanks to of the residual representation the construction of 푛-dimensional Galois representations attached to modular forms on unitary groups, itself ini- (2) 휚 ∶ 퐺 → GL (픽 ) ℚ,푆 2 ℓ tiated by Clozel and Kottwitz and then studied in great that is the reduction of 휚 modulo ℓ. In other words, one detail by Harris and Taylor in their proof of the local assumes at the outset the existence of a cuspidal modular Langlands conjectures for GL푛, allowing one to write the eigenform 푛-dimensional analogue of the map 푅휚,푆 → 핋휚,푆. 푛 The Taylor–Wiles method was generalized by Clozel, (3) 푓 = ∑ 푎푛(푓)푞 푛≥1 Harris, and Taylor in order to prove a modularity lifting which is matched with 휚, in the sense that for almost theorem in the minimal case. However, such a restriction on ramification makes these theorems very difficult to all primes 푝, the Fourier coefficient 푎푝(푓) is congruent apply in interesting situations. The general case was modulo ℓ to the integer 푎푝(퐸) = 푝 + 1 − #퐸(픽푝). This allows the introduction of a first key player, the Hecke treated using a generalization of the numerical criterion of Wiles, but only conditional on a conjecture (referred algebra 핋휚,푆, which acts faithfully on a space of cuspidal modular forms, all of whose Fourier coefficients agree to colloquially as Ihara’s lemma) that remains unproven modulo ℓ with those of 푓 and which have level supported today. at the primes of 푆. Kisin had earlier developed a generalization of the The second key player is the universal deformation ring Taylor–Wiles method in his study of modularity for GL2, in a work published in 2009. He enlarged the diagram (4) of 휚, which we call 푅휚,푆. It is a complete local ring which is characterized by a universal property. For example, the to a diagram set of homomorphisms 푅휚,푆 → ℤℓ is in bijection with the ̂ ′ (5) ⨂푝∈푆푅휚,푝 → 푅휚,푆 → 핋휚,푆, set of equivalence classes of lifts 휚 ∶ 퐺ℚ,푆 → GL2(ℤℓ) of 휚 that one hopes to prove are modular. where each ring 푅휚,푝 is an object parameterizing deforma- These key players are related by a surjective ring tions of the restriction of 휚 to a decomposition group at homomorphism 푝 (in other words, a local Galois group 퐷푝 = Gal(ℚ푝/ℚ푝)). In this point of view, the geometry of the rings 푅 begins (4) 푅휚,푆 → 핋휚,푆. 휚,푝 to play a key role. The Taylor–Wiles–Kisin method finally Jack Thorne is a Clay Research Fellow and Reader in Number allows one to link the modularity of Galois representa- Theory at the University of Cambridge. His e-mail address is tions 휚1, 휚2 ∶ 퐺ℚ,푆 → GL푛(ℤℓ) which have the property [email protected]. that the representations 휚1|퐷푝 , 휚2|퐷푝 determine points on

March 2017 Notices of the AMS 225 the same irreducible component of Spec 푅휚,푝 for each [2] Richard Taylor, Automorphy for some 푙-adic lifts of au- prime 푝 ∈ 푆. tomorphic mod 푙 Galois representations. II, Publ. Math. Inst. Building on this, Taylor [2] made a detailed study of Hautes Études Sci. No. 108 (2008), 183–239. MR2470688 the irreducible components of certain ‘local’ deformation (2010j:11085) [3] Richard Taylor and Andrew Wiles, Ring-theoretic proper- rings and introduced a very surprising trick that allowed ties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), him to circumvent completely the conjectural Ihara’s no. 3, 553–572. MR1333036 (96d:11072) lemma and prove the first modularity lifting theorems for GL푛 without restriction on the permitted ramification of Photo Credits 휚 relative to 휚. Photo taken at the Newton Institute is courtesy of Ken These theorems have had spectacular applica- Ribet. tions. Combined with the earlier work of Harris, Photo of Christopher Skinner is courtesy of William Crow/ Shepherd-Barron, and Taylor, they implied that the even- Princeton University. dimensional symmetric power Galois representations Photo of Barry Mazur is courtesy of Jim Harrison. 푛−1 (6) Sym 휚 ∶ 퐺ℚ,푆 → GL푛(ℤℓ) Photo of Mirela Çiperiani is by C. J. Mozzochi, courtesy of the Simons Foundation. are potentially modular: there exists a number field 퐹/ℚ 푛−1 Photo of Chandrashekhar Khare is by David Weisbart. such that the restriction Sym 휚|퐺퐹 to the absolute Galois group of 퐹 is modular, in the sense of being associated to modular (or automorphic) forms on GL푛,퐹. These ideas in turn led to the proof of the Sato–Tate conjecture for elliptic curves over ℚ: Theorem 1. Let 퐸 be an elliptic curve over ℚ without complex multiplication. Then the quantities 푎푝(퐸)/2√푝 ∈ [−1, 1] are equidistributed as 푝 → ∞ with respect to the Sato–Tate measure 2 (7) √1 − 푡2 푑푡. 휋 There are many fruitful directions that remain to be explored. For example, can one show that the symmetric powers (6) are modular and not just potentially modular? In joint work with Clozel [1], I showed that the answer to this question is affirmative for 푛 ≤ 9. A major part of our proof is a generalization of an important modularity lifting theorem of Skinner and Wiles which applies to Galois representations 휚 for which the residual representation 휚 is reducible. This modularity lifting theorem also played a major part in the proof of Serre’s conjecture. The proof of the Skinner–Wiles theo- The work of Wiles rem employs many ideas which appear in has had a Andrew Wiles’s most famous works, such transforming as 푝-adic families of modular forms and effect. congruences between Eisenstein series and cuspidal modular forms, as well as many other ideas whose importance would become apparent only later: we mention in particular an emphasis on the geometry of Galois deformation rings. The work of Wiles has had a transforming effect on this corner of number theory, and his influence continues to be felt throughout the subject.

References [1] Laurent Clozel and Jack A. Thorne, Level raising and sym- metric power functoriality, II, Ann. of Math. (2) 181 (2015), no. 1, 303–359. MR3272927

226 Notices of the AMS Volume 64, Number 3 ABOUT THE AUTHORS

Christopher Skinner studied at Karl Rubin was Andrew Wiles’s Princeton with Andrew Wiles in first PhD student. the mid-1990s.

Christopher Skinner Karl Rubin

Barry Mazur delights in the Mirela Çiperiani completed memories of the times Andrew her PhD in 2006 under the and he had at Harvard, as supervision of Andrew Wiles. colleagues.

Barry Mazur Mirela Çiperiani

Using techniques and ideas Jack Thorne studied at Harvard that grew out of Wiles’s proof with Richard Taylor, himself a of Fermat’s Last Theorem, student of Andrew Wiles. Chandrashekhar Khare (with Jean-Pierre Wintenberger) proved Serre’s Conjecture.

Chandrashekhar Jack Thorne Khare

March 2017 Notices of the AMS 227