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Wiles Wins the World's Oldest Scientific Prize Professor Sir Wiles WinsProfessor the World’s Sir Oldest Andrew Scientific Wiles Prize rofessor Sir Andrew Wiles of the University of Oxford Peter Swinnerton-Dyer observed a hitherto unexpected relation- is the 2017 recipient of the Copley Medal, the Royal Soci- ship between the algebraic and analytic aspects of elliptic curves. P ety’s oldest and most prestigious award. This honour cel- These curves typically have the form y2 = x3 + ax + b for con- ebrates his proof [1] of Fermat’s Last Theorem (FLT), and places stants a and b in a field K. Algebraically, one can impose an him in the company of such previous recipients as Gauss, Darwin, additive group structure on the K-rational points of the curve. and Einstein. In a modern practical application, the US National Security Agency recommends the use of the elliptic curve group for se- cure exchange of cryptographic keys. Analytically, each ellip- tic curve E is equipped with a so-called Hasse–Weil L-function, L(E,s). These L-functions are complex-valued functions of a complex variable, s, and generalise the celebrated Riemann zeta function. Birch and Swinnerton-Dyer conjectured that the rank Science Photo Library Science Photo | of the elliptic curve group is the order of the zero of L(E,s) at the point s =1. To date, both the Riemann hypothesis of 1859 and the Birch and Swinnerton-Dyer conjecture are open problems. A proof of Look At Sciences At Look | 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 © Frederic Woirgard © Frederic a special case of the Birch and Swinnerton-Dyer conjecture [4]. They showed that if E is an elliptic curve ‘with complex multipli- cation’ and if E has infinitely many points – equivalently, if the Wiles astounded the mathematical world in 1995 with a proof rank of E is positive – then L(E,1)=0and so the order of the of FLT that dominated seven years of his life, working in solitude zero of L(E,s) at s =1is also positive. Proving connections and secrecy. The result, first stated by Pierre de Fermat circa between seemingly unrelated mathematical entities is a hallmark 1630, is that for any integer n>2, there are no integers x, y, of Wiles’ work and would be the key to his proof of Fermat’s Last and z such that xn + yn = zn and xyz =0. The problem Theorem. stands in stark contrast to the equation x2 + y2 = z2 for which The story skips back now to post-war Japan, where Yutaka there are infinitely many essentially distinct solutions, known as Taniyama and Goro Shimura laid the foundation for Wiles’ fu- Pythagorean triples. Fermat claimed to have a proof of his Last ture success with a different insightful comparison, conjecturing Theorem, but whatever his argument was, it has been lost to the that every elliptic curve is modular. One description of modu- sands of time. larity involves the notion of modular forms of weight two and What survives from Fermat is a proof in the special case level N. These are functions defined on the upper half-plane where the exponent n =4. The proof is an early application H = z C : (z) > 0 that display a remarkable degree of { ∈ } of his idea of infinite descent, a variant of proof by contradic- symmetry. Among other properties, such a function f is invari- tion that proceeds by postulating the existence of a solution and ant under translation by 1 and satisfies derives from it a ‘smaller’ solution. The resulting infinite de- az + b creasing sequence of positive numbers is an absurd conclusion, f =(cz + d)2f(z), cz + d contradicting the assumption. Since Fermat’s time, countless other mathematicians have where a, b, c, d are integers with c divisible by N and with valiantly pitted their wits against the Herculean task of proving ad bc =1. his Last Theorem – every half-success resulting in breakthroughs − In number theory, modular forms have appeared in differ- in mathematical understanding. Indeed, the algebraic concept of ent guises over the years. The 19th century mathematician Carl an ideal arose in 1847 from an attempt by Ernst Kummer to fix Gustav Jacob Jacobi introduced theta functions to find the number Gabriel Lamé’s faulty proof of FLT [2]. It is well known that ev- of ways in which a natural number can be expressed as a sum of 4 ery whole number greater than 1 can be expressed uniquely as a squares. In modern language, his result rests upon the expression product of primes. Lamé’s attempted proof implicitly made this of modular forms as the Fourier series assumption, incorrectly, about other more general rings of alge- braic integers. Kummer restored uniqueness by using ‘ideal num- ∞ f(z)= 2πinz bers’ instead. In this sense, the story of Fermat’s Last Theorem bne . n=1 is the story of algebraic number theory. Wiles entered this story aged 10, when he found Fermat’s Last The other side of this coin concerns the arithmetic of elliptic Theorem stated in a library book. Though he lacked the tools at curves. For each prime p, one can reduce an elliptic curve modulo the time, his subsequent journey into more esoteric mathemat- p to produce a curve over a finite field. For all but finitely many ics would prove instrumental in chasing down his early quarry. primes, this curve is non-singular. The Shimura–Taniyama–Weil As a young researcher at Cambridge, he turned his hand to an- conjecture states that for each elliptic curve E, there is some mod- other famous problem, the conjecture of Birch and Swinnerton- ular form of weight 2 and level N such that for any prime p of Dyer [3]. In 1965, British mathematicians Bryan Birch and Sir good reduction, the number Np of points on the reduced curve Mathematics TODAY AUGUST 2017 142 satisfies N = p +1 b , where b is the Fourier coefficient Combined, these two statements imply the semi-stable Shimura– p − p p introduced above. What is more, the level N that appears in the Taniyama–Weil conjecture and hence Fermat’s last theorem. description of the modular form encodes information about the It is significant enough to prove such a long-standing conjec- primes of bad reduction, and equals an invariant called the con- ture. The continuing importance of Wiles’ work lies in the av- ductor of E. enues of research it has opened. Building on this work, the full The surprising link with FLT was conjectured by Gerhard Shimura–Taniyama–Weil conjecture – now known as the modu- Frey and proved by Ken Ribet in the mid 1980s. Frey asserted larity theorem – was proved [6] in 2001. Both results have given that FLT would follow from the Shimura–Taniyama–Weil conjec- impetus to Robert Langlands’ more general program for finding ture. His argument proceeds by contradiction. Suppose that FLT links between Galois representations and automorphic forms. It is false. Then there exists an odd prime p and non-zero integers is perhaps telling that Langlands and Wiles were both awarded the a, b, and c such that ap + bp = cp. Following Yves Hellegouarch, Wolf prize in 1996. Wiles’ Copley medal adds to a list of honours Frey considered the elliptic curve y2 = x(x ap)(x + bp). that include the Cole prize, the Abel prize, and a knighthood. − He believed that this curve could not be modular, contradict- Gihan Marasingha ing Shimura–Taniyama–Weil. However, his work was flawed. GihanUniversity Marasingha of Exeter Jean-Pierre Serre proposed that a weaker version of Shimura– University of Exeter Taniyama–Weil would suffice. He attempted to prove that FLT REFERENCES would follow if every semi-stable elliptic curve could be shown R1 Wiles, A. (1995) Modular elliptic curves and Fermat’s Last Theo- to be modular. Ken Ribet [5] filled the gap, humorously known rem, Ann. Math., vol. 142, pp. 443–551. 1 Wiles, A. (1995) Modular elliptic curves and Fermat’s Last Theorem, as the ε-conjecture, in Serre’s argument. 2 Ann.Stewart, Math. I., vol.and 142,Tall, pp. D. 443–551. (2002) Algebraic Number Theory and Fermat’s Last Theorem, 3rd edition, A.K. Peters, Ltd, Natick, MA. Upon hearing of Ribet’s result, Wiles’ childhood passion for 2 Stewart, I. and Tall, D. (2002) Algebraic Number Theory and Fermat’s FLT was rekindled. He set to work immediately trying to prove 3 LastBirch, Theorem B.J. and, 3rd Swinnerton-Dyer, edition, A.K. Peters, H.P.F. Ltd, (1965) Natick, Notes MA. on elliptic the semi-stable Shimura–Taniyama–Weil conjecture. His key re- 3 Birch,curves B.J.II, J. andReine Swinnerton-Dyer, Angew. Math., vol. H.P.F. 218, (1965) pp. 79–108. Notes on elliptic alisation was that elliptic curves could be studied indirectly via 4 curvesCoates, II, J.J. and Reine Wiles, Angew. A. Math.(1977), vol. On 218, the pp.conjecture 79–108. of Birch and their Galois representations. For each semi-stable elliptic curve 4 Coates,Swinnerton-Dyer, J. and Wiles, Invent. A. Math. (1977), vol. On 39, the pp. conjecture 223–251. of Birch and E and for each prime p, the Galois group Gal(Q¯ /Q) acts on the 5 Swinnerton-Dyer,Ribet, K. (1990) OnInvent. modular Math. representations, vol. 39, pp. 223–251. of Gal(Q/Q) arising group E[p] of p-torsion points of E. This action gives rise to a 5 Ribet,from modular K.
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