An astounding insight of Eichler was that for all prime numbers p, the coefficient bp is equal to ap. In other words, a2 = b2,a3 = b3,a5 = b5,a7 = b7, and so on.
Let’s check, for example, that this is true for p = 5. In this case, looking at the generating function we find that the coefficient in front of q5 is b5 = 1. On the other hand, we have seen that our cubic equation has 4 solutions modulo p = 5. Therefore a5 = 5 − 4 = 1, so indeed a5 = b5.
We started out with what looked like a problem of infinite complexity: counting solutions of the cubic equation
modulo p, for all primes p. And yet, all information about this problem is contained in a single line:
This one line is a secret code containing all information about the numbers of solutions of the cubic equation modulo all primes.
A useful analogy would be to think of the cubic equation like a sophisticated biological organism, and its solutions as various traits of this organism. We know that all of these traits are encoded in the DNA molecule. Likewise, all the complexity of our cubic equation turns out to be encoded in a generating function, which is like the DNA of this equation. Furthermore, this function is defined by a simple rule.
What’s even more fascinating is that if q is a number whose absolute value is less than 1, then the above infinite sum converges to a well-defined number. So we obtain a function in q, and this function turns out to have a very special property that is similar to the periodicity of the familiar trigonometric functions, sine and cosine.
The sine function sin(x) is periodic with the period 2π, that is to say, sin(x + 2π) = sin(x). But then also sin(x+4π) = sin(x), and more generally sin(x+2πn) = sin(x) for any integer n. Think about it this way: each integer n gives rise to a symmetry of the line: every point x on the line is shifted to x + 2πn. Therefore, the group of all integers is realized as a group of symmetries of the line. The periodicity of the sine function means that this function is invariant under this group.
Likewise, the Eichler generating function of the variable q written above turns out to be invariant under a certain symmetry group. Here we should take q to be not a real, but rather a complex number (we will discuss this topic in the next chapter). Then we can view q not as a point on the line, as in the case of the sine function, but as a point inside a unit disc on the complex plane. The symmetry property is similar: on this disc there is a group of symmetries, and our function is invariant under this group.12 A function with this kind of invariance property is called a modular form.
This symmetry group of the disc is very rich. To get an idea of what it is, let’s look at this picture, on which the disc is broken into infinitely many triangles.13
The symmetries act on the disc by exchanging these triangles. In fact, for any two triangles, there is a symmetry exchanging them. Though these symmetries of the disc are quite sophisticated, this is analogous to how, when the group of integers acts on the line, its symmetries move around the intervals [2πm,2π(m + 1)]. The sine function is invariant under those symmetries, whereas the Eichler generating function is invariant under symmetries of the disc.
As I mentioned at the beginning of this chapter, the sine function is the simplest example of a “harmonic” (basic wave) that is used in the harmonic analysis on the line. Likewise, the Eichler function, together with other modular forms, are the harmonics that appear in the harmonic analysis on the unit disc.
The magnificent insight of Eichler was that the seemingly random numbers of solutions of a cubic equation modulo primes come from a single generating function, which obeys an exquisite symmetry – revealing a hidden harmony and order in those numbers. Similarly, as if in a stroke of black magic, the Langlands Program organizes previously inaccessible information into regular patterns, weaving a delicate tapestry of numbers, symmetries, and equations.
When I first talked about mathematics at the beginning of this book, you may have wondered what I meant by a mathematical result being “beautiful” or “elegant.” This is what I meant. The fact that these highly abstract notions coalesce in such refined harmony is absolutely mind-boggling. It points to something rich and mysterious lurking beneath the surface, as if the curtain had been lifted and we caught glimpses of the reality that had been carefully hidden from us. These are the wonders of modern math, and of the modern world.
One might also ask whether, in addition to possessing innate beauty and establishing a surprising link between areas of mathematics that seem to be far removed from each other, this result has any practical applications. This is a fair question. At present, I am not aware of any. But cubic equations over finite fields of p elements of the kind we have considered above (which give rise to the so-called elliptic curves) are widely used in cryptography.14 So I would not be surprised if the analogues of Eichler’s result will also one day find applications as powerful and ubiquitous as encryption algorithms.
The Shimura–Taniyama–Weil conjecture is a generalization of Eichler’s result. It says that for any cubic equation like the one above (subject to some mild conditions), the numbers of solutions modulo primes are the coefficients of a modular form. Moreover, there is a one-to-one correspondence between the cubic equations and the modular forms of a certain kind.
What do I mean here by a one-to-one correspondence? Suppose that we have five pens and five pencils. We can assign a pencil to each pen in such a way that each pencil is assigned to one and only pen. This is called a one-to-one correspondence.
There are many different ways to do it. But suppose that under our one-to-one correspondence each pen has exactly the same length as the pencil assigned to it. We will then call the length an “invariant” and say that our correspondence preserves this invariant. If all pens have different lengths, the one-to-one correspondence will be uniquely determined by this property.
Now, in the case of the Shimura–Taniyama–Weil conjecture, the objects on one side are the cubic equations such as the one above. These will be our pens, and for each of them the numbers ap will be the invariants attached to it. (It’s like the length of a pen, except that now there isn’t just one invariant, but a whole collection labeled by primes p.)
The objects on the other side of the correspondence are modular forms. These will be our pencils, and for each of them, the coefficients bp will be the invariants attached to it (like the length of a pencil).
The Shimura–Taniyama–Weil conjecture says that there is a one-to-one correspondence between these objects preserving these invariants:
That is to say, for any cubic equation there exists a modular form such that ap = bp for all primes p, and vice versa.15
Now I can explain the link between the Shimura–Taniyama–Weil conjecture and Fermat’s Last Theorem: starting from a solution of the Fermat equation, we can construct a certain cubic equation.16 However, Ken Ribet showed that the numbers of solutions of this cubic equation modulo primes cannot be the coefficients of a modular form whose existence is stipulated by the Shimura–Taniyama–Weil conjecture. Once this conjecture is proved, we conclude that such a cubic equation cannot exist. Therefore, there are no solutions to the Fermat equation.
The Shimura–Taniyama–Weil conjecture is a stunning result because the numbers ap come from the study of solutions of an equation modulo primes – they are from the world of number theory – and the numbers bp are the coefficients of a modular form, from the world of harmonic analysis. These two worlds seem to be light years apart, and yet it turns out that they describe one and the same thing!
The Shimura–Taniyama–Weil conjecture may be recast as a special case of the Langlands Program. In order to do that, we replace each of the cubic equations appearing in the Shimura–Taniyama–Weil conjecture by a certain two-dimensional representation of the Galois group. This representation is naturally obtained from the cubic equation, and the numbers ap can be attached directly to this representation (rather than the cubic equation). Therefore the conjecture may be expressed as a relation between two-dimensional representati
ons of the Galois group and modular forms.
(I recall from Chapter 2 that a two-dimensional representation of a group is a rule that assigns a symmetry of a two-dimensional space (that is, a plane) to each element of this group. For example, in Chapter 2 we talked about a two-dimensional representation of the circle group.)
Even more generally, conjectures of the Langlands Program relate, in unexpected and profound ways, n-dimensional representations of the Galois group (which generalize the two-dimensional representations corresponding to the cubic equations in the Shimura–Taniyama–Weil conjecture) and the so-called automorphic functions (which generalize the modular forms in the Shimura–Taniyama–Weil conjecture):
Though there is little doubt that these conjectures are true, most of them are still unproved to this day, despite an enormous effort by several generations of mathematicians in the past forty-five years.
You may be wondering: how could one come up with these kinds of conjectures in the first place?
This is really a question about the nature of mathematical insight. The ability to see patterns and connections that no one had seen before does not come easily. It is usually the product of months, if not years, of hard work. Little by little, the inkling of a new phenomenon or a theory emerges, and at first you don’t believe it yourself. But then you say: “what if it’s true?” You try to test the idea by doing sample calculations. Sometimes these calculations are hard, and you have to navigate through mountains of heavy formulas. The probability of making a mistake is very high, and if it does not work at first, you try to redo it, over and over again.
More often than not, at the end of the day (or a month, or a year), you realize that your initial idea was wrong, and you have to try something else. These are the moments of frustration and despair. You feel that you have wasted an enormous amount of time, with nothing to show for it. This is hard to stomach. But you can never give up. You go back to the drawing board, you analyze more data, you learn from your previous mistakes, you try to come up with a better idea. And every once in a while, suddenly, your idea starts to work. It’s as if you had spent a fruitless day surfing, when you finally catch a wave: you try to hold on to it and ride it for as long as possible. At moments like this, you have to free your imagination and let the wave take you as far as it can. Even if the idea sounds totally crazy at first.
The statement of the Shimura–Taniyama–Weil conjecture must have sounded crazy to its creators. How could it not? Yes, the conjecture had its roots in earlier results, such as those by Eichler that we discussed above (which were subsequently generalized by Shimura), which showed that for some cubic equations, the numbers of solutions modulo p were recorded in the coefficients of a modular form. But the idea that this was true for any cubic equation must have sounded totally outrageous at the time. This was a leap of faith, first made by the Japanese mathematician Yutaka Taniyama, in the form of a question that he posed at the International Symposium on Algebraic Number Theory held in Tokyo in September 1955.
I’ve always wondered: what did it take for him to come to believe that this wasn’t crazy, but real? To have the courage to say it publicly?
We’ll never know. Unfortunately, not long after his great discovery, in November 1958, Taniyama committed suicide. He was only thirty-one. To add to the tragedy, shortly afterward the woman whom he was planning to marry also took her life, leaving the following note:17
We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him.
The conjecture was subsequently made more precise by Taniyama’s friend and colleague Goro Shimura, another Japanese mathematician. Shimura has worked most of his life at Princeton University and is currently an emeritus professor there. He has made major contributions to mathematics, many pertinent to the Langlands Program, and several fundamental concepts in this area carry his name (such as the “Eichler–Shimura congruence relations” and “Shimura varieties”).
In his thoughtful essay about Taniyama, Shimura made this striking comment:18
Though he was by no means a sloppy type, he was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this, and tried in vain to imitate him, but found it quite difficult to make good mistakes.
In the words of Shimura, Taniyama “wasn’t very careful when he stated his problem” at the Symposium in Tokyo in September 1955.19 Some corrections had to be made. And yet, this was a revolutionary insight, which led to one of the most significant achievements in mathematics of the twentieth century.
The third person whose name is attached to the conjecture is André Weil, whom I have mentioned earlier. He is one of the giants of mathematics in the twentieth century. Known for his brilliance as well as his temper, he was born in France and came to the United States during World War II. After holding academic appointments at various American universities, he settled at the Institute for Advanced Study in Princeton in 1958 and stayed there until his death in 1998, at age 92.
André Weil, 1981. Photo by Herman Landshoff. From the Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton.
Weil is particularly relevant to the Langlands Program, and not just because the famous letter in which Robert Langlands first formulated his ideas was addressed to him, or because of the Shimura–Taniyama–Weil conjecture. The Langlands Program is best seen through the prism of the “big picture” of mathematics that André Weil outlined in a letter to his sister. We will talk about it in the next chapter. This will be our stepping stone toward bringing the Langlands Program into the realm of geometry.
Chapter 9
Rosetta Stone
In 1940, during the war, André Weil was imprisoned in France for refusing to serve in the army. As the obituary published in The Economist put it,1
[Weil] had been deeply struck.... by the damage wreaked upon mathematics in France by the first world war, when “a misguided notion of equality in the face of sacrifice” led to the slaughter of the country’s young scientific elite. In the light of this, he believed he had a duty, not just to himself but also to civilization, to devote his life to mathematics. Indeed, he argued, to let himself be diverted from the subject would be a sin. When others raised the objection “but if everybody were to behave like you...”, he replied that this possibility seemed to him so implausible that he did not feel obliged to take it into account.
While in prison, Weil wrote a letter to his sister Simone Weil, a famous philosopher and humanist. This letter is a remarkable document; in it, he tries to explain in fairly elementary terms (accessible even to a philosopher – just kidding!) the “big picture” of mathematics as he saw it. Doing so, he set a great example to follow for all mathematicians. I sometimes joke that perhaps we should jail some of the leading mathematicians to force them to express their ideas in accessible terms, the way Weil did.
Weil writes in the letter about the role of analogy in mathematics, and he illustrates it by the analogy that interested him the most: between number theory and geometry.
This analogy proved to be extremely important for the development of the Langlands Program. As we discussed earlier, the roots of the Langlands Program are in number theory. Langlands conjectured that hard questions of number theory, such as the counting of solutions of equations modulo primes, can be solved by using methods of harmonic analysis – more specifically, the study of automorphic functions. This is exciting: first of all, it gives us a new way to solve what previously looked like intractable problems. And second, it points to deep and fundamental connections between different areas of mathematics. So naturally, we want to know what is really going on here: why might these hidden connections exist? And we still don’t fully understand it. Even the Shimura–Taniyama–Weil conjecture took a very long time to be resolved. And it’s only a special case of the general Langlands conjectures. There are hundreds and thousands of similar statements that are st
ill not proved.
So how should we approach these difficult conjectures? One way is just to keep working hard and try to come up with new ideas and insights. This has been happening, and significant progress has been made. Another possibility is to try to expand the scope of the Langlands Program. Since it points to some essential structures in number theory and harmonic analysis and connections between them, chances are that similar structures and connections can also be found between other fields of mathematics.
This has indeed turned out to be the case. It was gradually realized that the same mysterious patterns may be observed in other areas of mathematics, such as geometry, and even in quantum physics. When we learn something about these patterns in one area, we get hints about their meaning in other areas. I have written earlier that the Langlands Program is a Grand Unified Theory of mathematics. What I mean by this is that the Langlands Program points to some universal phenomena and connections between these phenomena across different fields of mathematics. And I believe that it holds the keys to understanding what mathematics is really about, far beyond the original Langlands conjectures.
The Langlands Program is now a vast subject. There is a large community of people working on it in different fields: number theory, harmonic analysis, geometry, representation theory, mathematical physics. Although they work with very different objects, they are all observing similar phenomena. And these phenomena give us clues to understanding how these diverse domains are interconnected, like parts of a giant jigsaw puzzle.
My entry point to the Langlands Program was through my work on Kac–Moody algebras, which I will describe in detail in the next few chapters. But the more I learned about the Langlands Program, the more I got excited by how ubiquitous it is in mathematics.
Love and Math Page 11