Professor Sir Andrew Wiles of the University of Oxford is the 2017 recipient of the Copley Medal, the Royal Society’s oldest and most prestigious award. This honour celebrates his proof  of Fermat’s Last Theorem (FLT), and places him in the company of such previous recipients as Gauss, Darwin, and Einstein.
Wiles astounded the mathematical world in 1995 with a proof of FLT that dominated seven years of his life, working in solitude and secrecy. The result, first stated by Pierre de Fermat circa 1630, is that for any integer , there are no integers , , and such that and . The problem stands in stark contrast to the equation for which there are infinitely many essentially distinct solutions, known as Pythagorean triples. Fermat claimed to have a proof of his Last Theorem, but whatever his argument was, it has been lost to the sands of time.
What survives from Fermat is a proof in the special case where the exponent . The proof is an early application of his idea of infinite descent, a variant of proof by contradiction that proceeds by postulating the existence of a solution and derives from it a ‘smaller’ solution. The resulting infinite decreasing sequence of positive numbers is an absurd conclusion, contradicting the assumption.
Since Fermat’s time, countless other mathematicians have valiantly pitted their wits against the Herculean task of proving his Last Theorem – every half-success resulting in breakthroughs in mathematical understanding. Indeed, the algebraic concept of an ideal arose in 1847 from an attempt by Ernst Kummer to fix Gabriel Lamé’s faulty proof of FLT . It is well known that every whole number greater than 1 can be expressed uniquely as a product of primes. Lamé’s attempted proof implicitly made this assumption, incorrectly, about other more general rings of algebraic integers. Kummer restored uniqueness by using ‘ideal numbers’ instead. In this sense, the story of Fermat’s Last Theorem is the story of algebraic number theory.
Wiles entered this story aged 10, when he found Fermat’s Last Theorem stated in a library book. Though he lacked the tools at the time, his subsequent journey into more esoteric mathematics would prove instrumental in chasing down his early quarry. As a young researcher at Cambridge, he turned his hand to another famous problem, the conjecture of Birch and Swinnerton-Dyer . In 1965, British mathematicians Bryan Birch and Sir Peter Swinnerton-Dyer observed a hitherto unexpected relationship between the algebraic and analytic aspects of elliptic curves. These curves typically have the form for constants and in a field . Algebraically, one can impose an additive group structure on the -rational points of the curve.
In a modern practical application, the US National Security Agency recommends the use of the elliptic curve group for secure exchange of cryptographic keys. Analytically, each elliptic curve is equipped with a so-called Hasse–Weil -function, . These -functions are complex-valued functions of a complex variable, , and generalise the celebrated Riemann zeta function. Birch and Swinnerton-Dyer conjectured that the rank of the elliptic curve group is the order of the zero of at the point .
To date, both the Riemann hypothesis of 1859 and the Birch and Swinnerton-Dyer conjecture are open problems. A proof of either of these conjectures would result in a $1 million prize from the Clay Mathematics Institute and, more importantly, everlasting fame in the annals of mathematics.
In 1977, Wiles and his supervisor, John Coates, established a special case of the Birch and Swinnerton-Dyer conjecture . They showed that if is an elliptic curve ‘with complex multiplication’ and if has infinitely many points – equivalently, if the rank of is positive – then and so the order of the zero of at is also positive. Proving connections between seemingly unrelated mathematical entities is a hallmark of Wiles’ work and would be the key to his proof of Fermat’s Last Theorem.
The story skips back now to post-war Japan, where Yutaka Taniyama and Goro Shimura laid the foundation for Wiles’ future success with a different insightful comparison, conjecturing that every elliptic curve is modular. One description of modularity involves the notion of modular forms of weight two and level . These are functions defined on the upper half-plane that display a remarkable degree of symmetry. Among other properties, such a function is invariant under translation by 1 and satisfies
where , , , are integers with divisible by and with .
In number theory, modular forms have appeared in different guises over the years. The 19th century mathematician Carl Gustav Jacob Jacobi introduced theta functions to find the number of ways in which a natural number can be expressed as a sum of 4 squares. In modern language, his result rests upon the expression of modular forms as the Fourier series
The other side of this coin concerns the arithmetic of elliptic curves. For each prime , one can reduce an elliptic curve modulo to produce a curve over a finite field. For all but finitely many primes, this curve is non-singular. The Shimura–Taniyama–Weil conjecture states that for each elliptic curve , there is some modular form of weight 2 and level such that for any prime of good reduction, the number of points on the reduced curve satisfies , where is the Fourier coefficient introduced above. What is more, the level that appears in the description of the modular form encodes information about the primes of bad reduction, and equals an invariant called the conductor of .
The surprising link with FLT was conjectured by Gerhard Frey and proved by Ken Ribet in the mid 1980s. Frey asserted that FLT would follow from the Shimura–Taniyama–Weil conjecture. His argument proceeds by contradiction. Suppose that FLT is false. Then there exists an odd prime and non-zero integers , , and such that . Following Yves Hellegouarch, Frey considered the elliptic curve . He believed that this curve could not be modular, contradicting Shimura–Taniyama–Weil. However, his work was flawed. Jean-Pierre Serre proposed that a weaker version of Shimura–Taniyama–Weil would suffice. He attempted to prove that FLT would follow if every semi-stable elliptic curve could be shown to be modular. Ken Ribet  filled the gap, humorously known as the -conjecture, in Serre’s argument.
Upon hearing of Ribet’s result, Wiles’ childhood passion for FLT was rekindled. He set to work immediately trying to prove the semi-stable Shimura–Taniyama–Weil conjecture. His key realisation was that elliptic curves could be studied indirectly via their Galois representations. For each semi-stable elliptic curve and for each prime , the Galois group acts on the group of -torsion points of . This action gives rise to a representation . Wiles proved that every such representation is ‘modular’. He also proved that if at least one representation of a semi-stable elliptic curve is modular, then so is the curve.
Combined, these two statements imply the semi-stable Shimura–Taniyama–Weil conjecture and hence Fermat’s last theorem.
It is significant enough to prove such a long-standing conjecture. The continuing importance of Wiles’ work lies in the avenues of research it has opened. Building on this work, the full Shimura–Taniyama–Weil conjecture – now known as the modularity theorem – was proved  in 2001. Both results have given impetus to Robert Langlands’ more general program for finding links between Galois representations and automorphic forms. It is perhaps telling that Langlands and Wiles were both awarded the Wolf prize in 1996. Wiles’ Copley medal adds to a list of honours that include the Cole prize, the Abel prize, and a knighthood.
University of Exeter
- Wiles, A. (1995) Modular elliptic curves and Fermat’s Last Theorem, Ann. Math., vol. 142, pp. 443–551.
- Stewart, I. and Tall, D. (2002) Algebraic Number Theory and Fermat’s Last Theorem, 3rd edition, A.K. Peters, Ltd, Natick, MA.
- Birch, B.J. and Swinnerton-Dyer, H.P.F. (1965) Notes on elliptic curves II, J. Reine Angew. Math., vol. 218, pp. 79–108.
- Coates, J. and Wiles, A. (1977) On the conjecture of Birch and Swinnerton-Dyer, Invent. Math., vol. 39, pp. 223–251.
- Ribet, K. (1990) On modular representations of arising from modular forms, Invent. Math., vol. 100, pp. 431–476.
- Breuil, C., Conrad, B., Diamond, F. and Taylor, R. (2001) On the modularity of elliptic curves over : wild 3-adic exercises, J. Amer. Math. Soc., vol. 14, pp. 843–939.
Reproduced from Mathematics Today, August 2017
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