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).