The solutions are:
Although some of these solutions would not be valid in normal arithmetic, in 5-clock arithmetic they are acceptable. For example, the fourth solution (x = 1, y = 4) works as follows:
But remember, 20 is equivalent to 0 in 5-clock arithmetic, because 5 will divide into 20 with a remainder of 0.
Because they could not list all the solutions to an elliptic equation working in infinite space, mathematicians, including Wiles, settled for working out the number of solutions in all the different clock arithmetics. For the elliptic equation given above the number of solutions in 5-clock arithmetic is four, and so mathematicians say E5 = 4. The number of solutions in other clock arithmetics can also be calculated. For example, in 7-clock arithmetic the number of solutions is nine, and so E7 = 9.
To summarise their results, mathematicians list the number of solutions in each clock arithmetic and call this list the L-series for the elliptic equation. What the L stands for has been long forgotten although some have suggested that it is the L of Gustav Lejeune-Dirichlet, who worked on elliptic equations. For clarity I will use the term E-series – the series that is derived from an elliptic equation. For the example given above the E-series is as follows:
Because mathematicians cannot say how many solutions some elliptic equations have in normal number space which extends up to infinity, the E-series appears to be next best thing. In fact the E-series encapsulates a great deal of information about the elliptic equation it describes. In the same way that biological DNA carries all the information required to construct a living organism, the E-series carries the essence of the elliptic equation. The hope was that by studying the E-series, this mathematical DNA, mathematicians would ultimately be able to calculate everything they could ever wish to know about an elliptic equation.
Working alongside John Coates, Wiles rapidly established his reputation as a brilliant number theorist with a profound understanding of elliptic equations and their E-series. As each new result was achieved and each paper published, Wiles did not realise that he was gathering the experience which would many years later bring him to the verge of a proof for Fermat’s Last Theorem.
Although nobody was aware of it at the time, the mathematicians of post-war Japan had already triggered a chain of events which would inextricably link elliptic equations to Fermat’s Last Theorem. By encouraging Wiles to study elliptic equations, Coates had given him the tools which would later enable him to work on his dream.
5
Proof by Contradiction
The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.
G.H. Hardy
In the January of 1954 a talented young mathematician at the University of Tokyo paid a routine visit to his departmental library. Goro Shimura was in search of a copy of Mathematische Annalen, Vol. 24. In particular he was after a paper by Deuring on his algebraic theory of complex multiplication, which he needed in order to help him with a particularly awkward and esoteric calculation.
To his surprise and dismay, the volume was already out. The borrower was Yutaka Taniyama, a vague acquaintance of Shimura who lived on the other side of the campus. Shimura wrote to Taniyama explaining that he urgently needed the journal to complete the nasty calculation, and politely asked when it would be returned.
A few days later, a postcard landed on Shimura’s desk. Taniyama had replied, saying that he too was working on the exact same calculation and was stuck at the same point in the logic. He suggested that they share their ideas and perhaps collaborate on the problem. This chance encounter over a library book ignited a partnership which would change the course of mathematical history.
Taniyama was born on 12 November 1927 in a small town a few miles north of Tokyo. The Japanese character symbolising his first name was intended to read ‘Toyo’, but most people outside his family misinterpreted it as ‘Yutaka’, and as Taniyama grew up he accepted and adopted this title. As a child Taniyama’s education was constantly interrupted. He suffered several bouts of ill health, and during his teenage years he was struck down by tuberculosis and had to miss two years of high school. The onset of war caused even greater disruption to his schooling.
Goro Shimura, one year younger than Taniyama, had his education stopped altogether during the war years. His school was shut down and, instead of attending lessons, Shimura had to help the war effort by working in a factory assembling aircraft parts. Each evening he would attempt to make up for his lost schooling and in particular found himself drawn to mathematics. ‘Of course there are many subjects to learn, but mathematics was the easiest because I could simply read mathematical textbooks. I learnt calculus by reading books. If I’d wanted to pursue chemistry or physics then I would have needed scientific equipment and I had no access to such things. I never thought that I was talented. I was just curious.’
A few years after the war had finished, Shimura and Taniyama found themselves at university. By the time they had exchanged postcards over the library book, life in Tokyo was beginning to return to normal and the two young academics could afford one or two small luxuries. They spent their afternoons in the coffee-shops, in the evenings they dined in a little restaurant specialising in whale meat, and at weekends they would stroll through the botanical gardens or the city park. All ideal locations for discussing their latest mathematical thoughts.
Although Shimura had a whimsical streak – even today he retains his fondness for Zen jokes – he was far more conservative and conventional than his intellectual partner. Shimura would rise at dawn and immediately get down to work, whereas his colleague would often still be awake at this time, having worked through the night. Visitors to his apartment would often find Taniyama fast asleep in the middle of the afternoon.
While Shimura was fastidious, Taniyama was sloppy to the point of laziness. Surprisingly this was a trait that Shimura admired: ‘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.’
Taniyama was the epitome of the absent-minded genius and this was reflected in his appearance. He was incapable of tying a decent knot, and so he decided that rather than tie his shoelaces a dozen times a day he would not tie them at all. He would always wear the same peculiar green suit with a strange metallic sheen. It was made from a fabric which was so outrageous that it had been rejected by the other members of his family.
When they met in 1954 Taniyama and Shimura were just beginning their mathematical careers. The tradition was, and still is, for young researchers to be taken under the wing of a professor who would guide the fledgling brain, but Taniyama and Shimura rejected this form of apprenticeship. During the war real research had ground to a halt and even by the 1950s the mathematics faculty had still not recovered. According to Shimura, the professors were ‘tired, jaded and disillusioned’. In comparison the post-war students were passionate and eager to learn, and they soon realised that the only way forward would be for them to teach themselves. The students organised regular seminars, taking it in turn to inform each other of the latest techniques and breakthroughs. Despite his otherwise lackadaisical attitude, when it came to the seminars Taniyama provided a ferocious driving force. He would encourage the more senior students to explore uncharted territory, and for the younger students he acted as a father figure.
Because of their isolation, the seminars would occasionally cover subjects which were generally considered passe in Europe and America. The students’ naïvety meant that they studied equations which had been abandoned in the West. One particularly unfashionable topic which fascinated both Taniyama and Shimura was the study of modular forms.
Modular forms are some of the weirdest and most wonderful objects in mathematics. They are one of th
e most esoteric entities in mathematics and yet the twentieth-century number theorist Martin Eichler rated them as one of the five fundamental operations: addition, subtraction, multiplication, division and modular forms. Most mathematicians would consider themselves masters of the first four operations, but the fifth one they still find a little confusing.
The key feature of modular forms is their inordinate level of symmetry. Although most people are familiar with the everyday concept of symmetry, it has a very particular meaning in mathematics, which is that an object has symmetry if it can be transformed in a particular way and yet afterwards appear to be unchanged. To appreciate the immense symmetry of a modular form it helps to first examine the symmetry of a more mundane object such as a simple square.
Figure 15. A simple square exhibits both rotational and reflectional symmetry.
In the case of a square, one form of symmetry is rotational. That is to say, if we imagine a pivot at the point where the x-axis and y-axis cross, then the square in Figure 15 can be rotated by one quarter of a turn, and afterwards it will appear to be unchanged. Similarly, rotations by half a turn, three-quarters of a turn and one full turn will also leave the square apparently unchanged.
In addition to rotational symmetry the square also possesses reflectional symmetry. If we imagine a mirror placed along the x-axis then the top half of the square would reflect exactly onto the lower half, and vice versa, so after the transformation the square would appear to remain unchanged. Similarly we can define three other mirrors (along the y-axis and along the two diagonals) for which the reflected square would appear to be identical to the original one.
The simple square is relatively symmetric, possessing both rotational and reflectional symmetries, but it does not possess any translational symmetry. This means that if the square were to be shifted in any direction, an observer would spot the movement immediately because its position relative to the axes would have changed. However, if the whole of the space were tiled with squares, as shown in Figure 16, this infinite collection of squares would then have translational symmetry. If the infinite tiled surface were to be shifted up or down by one or more tile spaces, then the translated tiling would appear to be identical to the original one.
Figure 16. An infinite surface tiled with squares exhibits rotational and reflectional symmetry, and in addition has translational symmetry.
The symmetry of tiled surfaces is a relatively straightforward idea, but as with many seemingly simple concepts there are many subtleties hidden within it. For example, in the 1970s the British physicist and recreational mathematician Roger Penrose began dabbling with different tiles on the same surface. Eventually he identified two particularly interesting shapes, called the kite and the dart, which are shown in Figure 17. On their own, there is only one way these shapes can be used to tile a surface without gaps or overlaps, but together they could be used to create a rich set of tiling patterns. The kites and darts can be fitted together in an infinite number of ways, and although each pattern is apparently similar, in detail they all vary. One pattern made from kites and darts is shown in Figure 17.
Figure 17. By using two different tiles, the kite and the dart, Roger Penrose was able to cover a surface. However, Penrose tiling does not possess translational symmetry.
Another remarkable feature of Penrose tilings (the patterns generated by tiles such as the kite and dart) is that they can exhibit a very restricted level of symmetry. At first sight it would appear that the tiling shown in Figure 17 would have translational symmetry, and yet any attempt to shift the pattern across so that it effectively remains unchanged ends in failure. Penrose tilings are deceptively unsymmetrical, and this is why they fascinate mathematicians and have become the starting point for a whole new area of mathematics.
Curiously Penrose tiling has also had repercussions in material science. Crystallographers always believed that crystals had to be built on the principles behind square tiling, possessing a high level of translational symmetry. In theory building crystals relied on a highly regular and repetitive structure. However, in 1984 scientists discovered a metallic crystal made of aluminium and manganese which was built along Penrose principles. The mosaic of aluminium and manganese behaved like the kites and darts, generating a crystal which was almost regular, but not quite. A French company has recently developed a Penrose crystal into a coating for frying-pans.
While the fascinating thing about Penrose’s tiled surfaces is their restricted symmetry, the interesting property of modular forms is that they exhibit infinite symmetry. The modular forms studied by Taniyama and Shimura can be shifted, switched, swapped, reflected and rotated in an infinite number of ways and still they remain unchanged, making them the most symmetrical of mathematical objects. When the French polymath Henri Poincaré studied modular forms in the nineteenth century, he had great difficulty coming to terms with their immense symmetry. After working on a particular type of modular form, he described to his colleagues how every day for two weeks he would wake up and try and find an error in his calculations. On the fifteenth day he realised and accepted that modular forms were indeed symmetrical in the extreme.
Unfortunately, drawing, or even imagining, a modular form is impossible. In the case of the square tiling we have an object which lives in two dimensions, its space being defined by the x-axis and the y-axis. A modular form is also defined by two axes, but the axes are both complex, i.e. each axis has a real and an imaginary part and effectively becomes two axes. Therefore the first complex axis must be represented by two axes, xr-axis (real) and xi-axis (imaginary), and the second complex axis is represented by two axes, yr-axis (real) and yi-axis (imaginary). To be precise, modular forms live in the upper half-plane of this complex space, but what is most important to appreciate is that this is a four-dimensional space (xr, xi, yr, yi).
This four-dimensional space is called hyperbolic space. The hyperbolic universe is tricky to comprehend for humans, who are constrained to living in a conventional three-dimensional world, but four-dimensional space is a mathematically valid concept, and it is this extra dimension which gives the modular forms such an immensely high level of symmetry. The artist Mauritz Escher was fascinated by mathematical ideas and attempted to convey the concept of hyperbolic space in some of his etchings and paintings. Escher’s Circle Limit IV embeds the hyperbolic world into the two-dimensional page. In true hyperbolic space the devils and angels would be the same size, and the repetition is indicative of the high level of symmetry. Although some of this symmetry can be seen on the two-dimensional page, there is an increasing distortion towards the edge of the picture.
The modular forms which live in hyperbolic space come in various shapes and sizes, but each one is built from the same basic ingredients. What differentiates each modular form is the amount of each ingredient it contains. The ingredients of a modular form are labelled from one to infinity (M1, M2, M3, M4, …) and so a particular modular form might contain one lot of ingredient one (M1 = 1), three lots of ingredient two (M2 = 3), two lots of ingredient three (M3 = 2), etc. This information describing how a modular form is constructed can be summarised in a so-called modular series, or M-series, a list of the ingredients and the quantity of each one required:
Just as the E-series is the DNA for elliptic equations, the M-series is the DNA for modular forms. The amount of each ingredient listed in the M-series is critical. Depending how you change the amount of, say, the first ingredient you might generate a completely different, but equally symmetrical, modular form, or you might destroy the symmetry altogether and generate a new object which is not a modular form. If the quantity of each ingredient is arbitrarily chosen, then the result will probably be an object with little or no symmetry.
Modular forms stand very much on their own within mathematics. In particular, they would seem to be completely unrelated to the subject that Wiles would study at Cambridge, elliptic equations. The modular form is an enormously complicated beast, studied largely because of its symm
etry and only discovered in the nineteenth century. The elliptic equation dates back to the ancient Greeks and has nothing to do with symmetry. Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects. However, Taniyama and Shimura were to shock the mathematical community by suggesting that elliptic equations and modular forms were effectively one and the same thing. According to these two maverick mathematicians, they could unify the modular and elliptic worlds.
Wishful Thinking
In September 1955 an international symposium was held in Tokyo. It was a unique opportunity for the many young Japanese researchers to show off to the rest of the world what they had learned. They handed round a collection of thirty-six problems related to their work, accompanied by a humble introduction – Some unsolved problems in mathematics: no mature preparation has been made, so there may be some trivial or already solved ones among these. The participants are requested to give comments on any of these problems.
Four of the questions were from Taniyama, and these hinted at a curious relationship between modular forms and elliptic equations. These innocent questions would ultimately lead to a revolution in number theory. Taniyama had looked at the first few terms in the M-series of a particular modular form. He recognised the pattern and realised that it was identical to the list of numbers in the E-series of a well-known elliptic equation. He calculated a few more terms in each series, and still the M-series of the modular form and E-series of the elliptic equation matched perfectly.