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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 ’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 . According to the prize citation, Wiles merits this recogni- tion “for his stunning proof of Fermat’s Last Theorem by way of the modularity for semistable elliptic curves, opening a new era in .” Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the 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 ’s contribution in this volume). the shelves of imposing, ambitious It is also a centerpiece of the “,” the his local pub- imposing, ambitious edifice of results and 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 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 (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 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 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 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 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).

210 Notices of the AMS Volume 64, Number 3 Moving on to equations of degree two, the general involves the same basic constituents, Dirichlet series, quadratic equation in one variable is of the form 푎푥2 +푏푥+ as in the one-variable case. This means that quadratic 푐 = 0, and its behaviour is governed by its discriminant diophantine equations in any number of variables are well understood, at least as far as their zeta functions are (15) Δ ∶= 푏2 − 4푎푐. concerned. This purely algebraic fact remains true over the finite The plot thickens when equations of higher degree fields, and for primes 푝 ∤ 2푎Δ one has are considered. Consider for instance the cubic equation 3 0 if Δ is a nonsquare modulo 푝, 푥 − 푥 − 1 of discriminant Δ = −23. For all 푝 ≠ 23, (16) 푁 = { 푝 2 if Δ is a square modulo 푝. this cubic equation has no multiple roots over 픽푝푟 , and therefore 푁푝 = 0, 1, or 3. A simple expression for 푁푝 in A priori, the criterion for whether 푁푝 = 2 or 0—whether this case is given by the following theorem of Hecke: the integer Δ is or is not a quadratic residue modulo 푝—seems like a subtle condition on the prime 푝. To get Theorem (Hecke). The following hold for all primes 푝 ≠ a better feeling for this condition, consider the example 23: 2 of the equation 푥 − 푥 − 1, for which Δ = 5. Calculating (1) If 푝 is not a square modulo 23, then 푁푝 = 1. whether 5 is a square or not modulo 푝 for the first few (2) If 푝 is a square modulo 23, then primes 푝 ≤ 101 leads to the following list: 0 if 푝 = 2푎2 + 푎푏 + 3푏2, (17) (21) 푁 = { 푝 3 if 푝 = 푎2 + 푎푏 + 6푏2, 2 for 푝=11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, … 푁 ={ 푝 0 for 푝=2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, … for some 푎, 푏 ∈ ℤ. A clear pattern emerges from this experiment: whether Hecke’s theorem implies that 푁 = 0 or 2 seems to depend only on the rightmost digit ∞ 푝 3 −푠 of 푝, i.e., on what the remainder of 푝 is modulo 10. One (22) 휁(푥 − 푥 − 1; 푠) = 휁(푠) × ∑ 푎푛푛 , is led to surmise that 푛=1 where the generating series 2 if 푝 ≡ 1, 4 (mod 5), (18) 푁푝 = { (23) 0 if 푝 ≡ 2, 3 (mod 5), 푛 2 3 6 8 13 16 23 퐹(푞)∶=∑ 푎푛푞 = 푞−푞 −푞 +푞 +푞 −푞 −푞 +푞 +⋯ a formula that represents a dramatic improvement over is given by the explicit formula (16), allowing a much more efficient calculation of 푁푝 for example. The guess in (18) is in fact a consequence of 1 2 2 2 2 (24) 퐹(푞) = ⎛ ∑ 푞푎 +푎푏+6푏 − 푞2푎 +푎푏+3푏 ⎞ . Gauss’s celebrated law of quadratic reciprocity: 2 ⎝푎,푏∈ℤ ⎠ 2 Theorem (Quadratic reciprocity). For any equation 푎푥 + The function 푓(푧) = 퐹(푒2휋푖푧) that arises by setting 푞 = 2 푏푥+푐, with Δ ∶= 푏 −4푎푐, the value of the function 푝 ↦ 푁푝 푒2휋푖푧 in (24) is a prototypical example of a : (for 푝 ∤ 푎Δ) depends only on the residue class of 푝 modulo namely, it satisfies the transformation rule 4Δ and hence is periodic with period length dividing 4|Δ|. (25) 푎푧 + 푏 푎, 푏, 푐, 푑 ∈ ℤ, 푎푑 − 푏푐 = 1 The repeating pattern satisfied by the 푁푝’s as 푝 varies 푓 ( ) = (푐푧+푑)푓(푧), { 푎 greatly facilitates the manipulation of the zeta functions 푐푧 + 푑 23|푐, ( 23 ) = 1. of quadratic equations. For example, the zeta function of under so-called modular substitutions of the form 푧 ↦ 푎푧+푏 . 풳 ∶ 푥2 − 푥 − 1 = 0 is equal to 푐푧+푑 This property follows from the Poisson summation for- 1 1 1 1 1 1 1 휁(풳; 푠) = 휁(푠)×{(1− − + )+( − − + ) mula applied to the expression in (24). Thanks to (25), the 2푠 3푠 4푠 6푠 7푠 8푠 9푠 zeta function of 풳 can be manipulated with the same ease 1 1 1 1 as the zeta functions of Riemann and Dirichlet. Indeed, (19) + (1 − − + )+⋯}. 푠 푠 푠 푠 ∞ −푠 11 12 13 14 Hecke showed that the 퐿-series ∑푛=1 푎푛푛 attached to ∞ 2휋푖푛푧 The series that occurs on the right-hand side is a prototyp- a modular form ∑푛=1 푎푛푒 possesses very similar an- ical example of a Dirichlet 퐿-series. These 퐿-series, which alytic properties, notably an analytic continuation and a are the key actors in the proof of Dirichlet’s theorem on Riemann-style functional equation. the infinitude of primes in arithmetic progressions, enjoy The generating series 퐹(푞) can also be expressed as an many of the same analytic properties as the Riemann zeta infinite product: function: an analytic continuation to the entire complex (26) ∞ plane and a functional equation relating their values at 푠 1 2 2 2 2 ⎛ 푞푎 +푎푏+6푏 −푞2푎 +푎푏+3푏 ⎞=푞 (1−푞푛)(1−푞23푛). and 1 − 푠. They are also expected to satisfy a Riemann hy- ∑ ∏ 2 ⎝푎,푏∈ℤ ⎠ 푛=1 pothesis which generalises Riemann’s original statement and is just as deep and elusive. The first few terms of this power series identity can readily It is a (not completely trivial) fact that the zeta function be verified numerically, but its proof is highly nonobvious of the general quadratic equation in 푛 variables and indirect. It exploits the circumstance that the space of holomorphic functions of 푧 satisfying the transformation 푛 푛 rules (25) together with suitable growth properties is a one- (20) ∑ 푎푖푗푥푖푥푗 + ∑ 푏푖푥푖 + 푐 = 0 푖,푗=1 푖=1 dimensional complex vector space which also contains

March 2017 Notices of the AMS 211 the infinite product above. Indeed, the latter is equal to after a suitable change of variables and which are non- 휂(푞)휂(푞23), where singular, a condition equivalent to the assertion that the 3 2 ∞ discriminant Δ ∶= −16(4푎 + 27푏 ) is nonzero. (27) 휂(푞) = 푞1/24 ∏(1 − 푞푛) To illustrate Wiles’s theorem with a concrete example, 푛=1 consider the equation is the Dedekind eta function whose logarithmic derivative (32) 퐸 ∶ 푦2 = 푥3 − 푥, (after viewing 휂 as a function of 푧 through the change of 2휋푖푧 of discriminant Δ = 64. After setting variables 푞 = 푒 ) is given by (33) ∞ −푠 −푠 −푠 −1 휂′(푧) −1 휁(퐸; 푠) = 휁(푠−1)×(푎1 + 푎22 + 푎33 + 푎44 + ⋯) , (28) = −휋푖( + 2 ∑ ( ∑ 푑)푒2휋푖푛푧) 휂(푧) 12 푛=1 푑|푛 the associated generating series satisfies the following identities reminiscent of (24) and (26): 푖 ∞ ∞ 1 (29) = , (34) ∑ ∑ 2 4휋 푚=−∞ 푛=−∞ (푚푧 + 푛) 푛 5 9 13 17 25 where the term attached to (푚, 푛) = (0, 0) is excluded 퐹(푞) = ∑ 푎푛푞 = 푞 − 2푞 − 3푞 + 6푞 + 2푞 − 푞 + ⋯ 2 2 from the last sum. The Dedekind 휂-function is also (35) = ∑ 푎 ⋅ 푞(푎 +푏 ) connected to the generating series for the partition 푎,푏 function 푝(푛) describing the number of ways in which ∞ 푛 can be expressed as a sum of positive via the (36) = 푞 ∏(1 − 푞4푛)2(1 − 푞8푛)2, identity 푛=1 2 ∞ where the sum in (35) runs over the (푎, 푏) ∈ ℤ for which (30) 휂−1(푞) = 푞−1/24 ∑ 푝(푛)푞푛, the Gaussian integer 푎 + 푏푖 is congruent to 1 modulo 푛=0 (1+푖)3. (This identity follows from Deuring’s study of zeta which plays a starring functions of elliptic curves with role alongside Jeremy and may even have been known earlier.) Once again, the 2휋푖푧 “There are five Irons and Dev Patel in holomorphic function 푓(푧) ∶= 퐹(푒 ) is a modular form a recent film about satisfying the slightly different transformation rule elementary the life of Srinivasa (37) 푎푧 + 푏 푎, 푏, 푐, 푑 ∈ ℤ, 푎푑 − 푏푐 = 1, Ramanujan. 푓 ( ) = (푐푧+푑)2푓(푧), { arithmetical Commenting on the 푐푧 + 푑 32|푐. operations: “unreasonable effec- Note the exponent 2 that appears in this formula. Because tiveness and ubiquity of it, the function 푓(푧) is said to be a modular form of addition, of modular forms,” weight 2 and level 32. The modular forms of (25) attached Martin Eichler once to cubic equations in one variable are of weight 1, but subtraction, wrote, “There are otherwise the parallel of (35) and (36) with (24) and (26) five elementary arith- is striking. The original conjecture of Shimura–Taniyama, multiplication, metical operations: and Weil asserts the same pattern for all elliptic curves: addition, subtraction, division,…and multiplication, divi- Conjecture (Shimura–Taniyama–Weil). Let 퐸 be any ellip- sion,…and modular tic curve. Then modular forms.” ∞ −1 forms.” Equations (26), (38) 휁(퐸; 푠) = 휁(푠 − 1) × ( 푎 푛−푠) , (29), and (30) are just a ∑ 푛 푛=1 few of the many won- 2휋푖푛푧 drous identities which abound, like exotic strains of where 푓퐸(푧) ∶= ∑ 푎푛푒 is a modular form of weight 2. fragrant wild orchids, in what Roger Godement has called The conjecture was actually more precise and predicted the “garden of modular delights.” that the level of 푓퐸—i.e., the integer that appears in the The example above and many others of a similar type transformation property for 푓퐸 as the integers 23 and 32 are described in Jean-Pierre Serre’s delightful monograph in (25) and (37) respectively—is equal to the arithmetic [Se], touching on themes that were also covered in Serre’s conductor of 퐸. This conductor, which is divisible only by lecture at the inaugural Abel Prize ceremony in 2003. primes for which the equation defining 퐸 becomes singular Hecke was able to establish that all cubic polynomials in modulo 푝, is a measure of the arithmetic complexity of one variable are modular; i.e., the coefficients of their zeta 퐸 and can be calculated explicitly from an equation for functions obey patterns just like those of (24) and (25). 퐸 by an algorithm of Tate. An elliptic curve is said to be Wiles’s achievement was to extend this result to a large semistable if its arithmetic conductor is squarefree. This class of cubic diophantine equations in two variables over class of elliptic curves includes those of the form the rational numbers: the elliptic curve equations which (39) 푦2 = 푥(푥 − 푎)(푥 − 푏) can be brought to the form with gcd(푎, 푏) = 1 and 16|푏. The most famous elliptic (31) 푦2 = 푥3 + 푎푥 + 푏 curves in this class are those that ultimately do not exist:

212 Notices of the AMS Volume 64, Number 3 compact subgroup GL2(ℤ푝) for all but finitely many 푝). The shift in emphasis from modular forms to the so-called automorphic representations which they span is decisive. Langlands showed how to attach an 퐿-function to any irreducible automorphic representation of 퐺(픸ℚ) for an arbitrary reductive algebraic group 퐺, of which the matrix groups GL푛 and more general algebraic groups of Lie type are prototypical examples. This greatly enlarges the notion of what it means to be “modular”: a diophantine equation is now said to have this property if its zeta function can be expressed in terms of the Langlands 퐿- functions attached to automorphic representations. One of the fundamental goals in the Langlands program is to establish further cases of the following conjecture: Conjecture. All diophantine equations are modular in the above sense. This conjecture can be viewed as a far-reaching gen- eralisation of quadratic reciprocity and underlies the non-Abelian reciprocity laws that are at the heart of ’s achievement. Before Wiles’s proof, the following general classes of diophantine equations were known to be modular: Andrew Wiles, Henri Darmon, and Mirela Çiperiani in • Quadratic equations, by Gauss’s law of quadratic June 2016 at Harvard University during a conference reciprocity; in honor of Karl Rubin’s sixtieth birthday. • Cubic equations in one variable, by the work of Hecke and Maass; the “Frey curves” 푦2 = 푥(푥 − 푎푝)(푥 + 푏푝) arising from • Quartic equations in one variable. 푝 푝 푝 putative solutions to Fermat’s equation 푎 + 푏 = 푐 , This last case deserves further comment, since it has whose nonexistence had previously been established in a not been discussed previously and plays a primordial 1 landmark article of Kenneth Ribet under the assumption role in Wiles’s proof. The modularity of quartic equations of their modularity. It is the proof of the Shimura– follows from the seminal work of Langlands and Tunnell. Taniyama–Weil conjecture for semistable elliptic curves While it is beyond the scope of this survey to describe that earned Andrew Wiles the Abel Prize: their methods, it must be emphasised that Langlands and Theorem (Wiles). Let 퐸 be a semistable elliptic curve. Then Tunnell make essential use of the solvability by radicals of the general quartic equation, whose underlying symmetry 퐸 satisfies the Shimura–Taniyama–Weil conjecture. group is contained in the permutation group 푆4 on 4 letters. The semistability assumption in Wiles’s theorem was Solvable extensions are obtained from a succession of later removed by , , Fred Abelian extensions, which fall within the purview of the Diamond, and Richard Taylor around 1999. (See, for class field theory developed in the nineteenth and first instance, the account [Da] that appeared in the Notices at half of the twentieth century. On the other hand, the the time.) modularity of the general equation of degree > 4 in one As a prelude to describing some of the important ideas variable, which cannot be solved by radicals, seemed to in its proof, one must first try to explain why Wiles’s lie well beyond the scope of the techniques that were theorem occupies such a central position in mathematics. available in the “pre-Wiles era.” The reader who perseveres The Langlands program places it in a larger context to the end of this essay will be given a glimpse of how by offering a vast generalisation of what it means for our knowledge of the modularity of the general quintic a diophantine equation to be “associated to a modular equation has progressed dramatically in the wake of form.” The key is to view modular forms attached to cubic Wiles’s breakthrough. equations or to elliptic curves as in (24) or (34) as vectors Prior to Wiles’s proof, modularity was also not known in certain irreducible infinite-dimensional representations for any interesting general class of equations (of degree of the locally compact topological group > 2, say) in more than one variable; in particular it had only been verified for finitely many elliptic curves over ℚ (40) GL (픸 ) = ′ GL (ℚ ) × GL (ℝ) 2 ℚ ∏ 푝 2 푝 2 up to isomorphism over ℚ̄ (including the elliptic curves ′ (where ∏푝 denotes a restricted direct product over all over ℚ with complex multiplication, of which the elliptic the prime numbers consisting of elements (훾푝)푝 for curve of (31) is an instance.) Wiles’s modularity theorem which the 푝th component 훾푝 belongs to the maximal confirmed the Langlands conjectures in the important test case of elliptic curves, which may seem like (and, in 1See the interview with Ribet as the new AMS president in this fact, are) very special diophantine equations, but have issue, page 229. provided a fertile terrain for arithmetic investigations,

March 2017 Notices of the AMS 213 both in theory and in applications (cryptography, coding Like diophantine equations, Galois representations also theory…). give rise to analogous zeta functions. More precisely, the Returning to the main group 퐺ℚ,푆 contains, for each prime 푝 ∉ 푆, a distinguished theme of this report, element called the Frobenius element at 푝, denoted 휎푝. Wiles’s proof is Wiles’s proof is also impor- Strictly speaking, this element is well defined only up tant for having introduced to conjugacy in 퐺ℚ,푆, but this is enough to make the also important a revolutionary new ap- arithmetic sequence proach which has opened 푟 (42) 푁푝푟 (휚) ∶= Trace(휚(휎푝 )) for having the floodgates for many well defined. The zeta function 휁(휚; 푠) packages the further breakthroughs in information from this sequence in exactly the same way introduced a the Langlands program. as in the definition of 휁(풳; 푠). To expand on this revolutionary For example, if 풳 is attached to a polynomial 푃 of point, we need to present degree 푑 in one variable, the action of 퐺 on the another of the dramatis ℚ,푆 new approach roots of 푃 gives rise to a 푑-dimensional permutation personae in Wiles’s proof: representation which has Galois representations. Let ̄ (43) 휚 ∶ 퐺 ⟶ GL (ℚ), opened the 퐺ℚ = Gal(ℚ/ℚ) be the 풳 ℚ,푆 푑 absolute Galois group of and 휁(풳, 푠) = 휁(휚풳, 푠). This connection goes far deeper, floodgates for ℚ, namely, the automor- extending to diophantine equations in 푛 + 1 variables for phism group of the field general 푛 ≥ 0. The glorious insight at the origin of the many further of all algebraic numbers. Weil conjectures is that 휁(풳; 푠) can be expressed in terms It is a profinite group, of the zeta functions of Galois representations arising breakthroughs endowed with a natu- in the étale cohomology of 풳, a cohomology theory with ral topology for which ℓ-adic coefficients which associates to 풳 a collection in the Langlands the subgroups Gal(ℚ/퐿)̄ 푖 {퐻et(풳/ℚ̄ ,ℚℓ)} with 퐿 ranging over the 0≤푖≤2푛 program. finite extensions of ℚ of finite-dimensional ℚℓ-vector spaces endowed with a form a basis of open continuous linear action of 퐺ℚ,푆. (Here 푆 is the set of subgroups. Following the primes 푞 consisting of ℓ and the primes for which the original point of view taken by Galois himself, the group equation of 풳 becomes singular after being reduced 퐺ℚ acts naturally as permutations on the roots of poly- modulo 푞.) These representations generalise the repre- nomials with rational coefficients. Given a finite set 푆 of sentation 휚풳 of (43), insofar as the latter is realised by 0 primes, one may consider only the monic polynomials the action of 퐺ℚ,푆 on 퐻et(풳ℚ̄ ,ℚℓ) after extending the with integer coefficients whose discriminant is divisible coefficients from ℚ to ℚℓ. only by primes ℓ ∈ 푆 (eventually after a change of vari- Theorem (Weil, Grothendieck,…). If 풳 is a diophantine ables). The topological group 퐺ℚ operates on the roots of equation having good reduction outside 푆, there exist Ga- such polynomials through a quotient, denoted 퐺ℚ,푆—the lois representations 휚1 and 휚2 of 퐺ℚ,푆 for which automorphism group of the maximal algebraic extension unramified outside 푆, which can be regarded as the sym- (44) 휁(풳; 푠) = 휁(휚1; 푠)/휁(휚2; 푠). 푖 metry group of all the zero-dimensional varieties over ℚ The representations 휚1 and 휚2 occur in ⊕퐻et(풳/ℚ̄ ,ℚℓ), having “nonsingular reduction outside 푆.” where the direct sum ranges over the odd and even values In addition to the permutation representations of 퐺ℚ of 0 ≤ 푖 ≤ 2푛 for 휚1 and 휚2 respectively. More canonically, that were so essential in Galois’s original formulation there are always irreducible representations 휚1, … , 휚푡 of of his theory, it has become important to study the 퐺ℚ,푠 and integers 푑1, … 푑푡 such that (continuous) linear representations 푡 푑푖 (45) 휁(풳; 푠) = ∏ 휁(휚푖; 푠) , (41) 휚 ∶ 퐺ℚ,푆 ⟶ 퐺퐿푛(퐿) 푖=1 of this Galois group, where 퐿 is a complete field, such as arising from the decompositions of the (semisimplifi- the fields ℝ or ℂ of real or complex numbers, the finite 푖 cation of the) 퐻et(풳ℚ̄ ,ℚℓ) into a sum of irreducible field 픽ℓ푟 equipped with the discrete topology, or a finite representations. The 휁(휚푖, 푠) can be viewed as the ̄ extension 퐿 ⊂ ℚℓ of the field ℚℓ of ℓ-adic numbers. “atomic constituents” of 휁(풳, 푠) and reveal much of Galois representations were an important theme in the the “hidden structure” in the underlying equation. The de- work of Abel and remain central in modern times. Many composition of 휁(풳; 푠) into a product of different 휁(휚푖; 푠) illustrious mathematicians in the twentieth century have is not unlike the decomposition of a wave function into contributed to their study, including three former Abel its simple harmonics. Prize winners: Jean-Pierre Serre, , and Pierre A Galois representation is said to be modular if its zeta Deligne. Working on Galois representations might seem function can be expressed in terms of generating series to be a prerequisite for an algebraic number theorist to attached to modular forms and automorphic representa- receive the Abel Prize! tions and is said to be geometric if it can be realised in

214 Notices of the AMS Volume 64, Number 3 an étale cohomology group of a diophantine equation as the inverse limit being taken relative to the multiplication- 푛 above. The “main conjecture of the Langlands program” by-3 maps. The groups 퐸[3 ] and 푇3(퐸) are free modules 푛 can now be amended as follows: of rank 2 over (ℤ/3 ℤ) and ℤ3 respectively and are endowed with continuous linear actions of 퐺 , where 푆 Conjecture. All geometric Galois representations of 퐺 ℚ,푆 ℚ,푆 is a set of primes containing 3 and the primes that divide are modular. the conductor of 퐸. One obtains the associated Galois Given a Galois representation representations:

(46) 휚 ∶ 퐺ℚ,푆 ⟶ GL푛(ℤ ) 휚퐸,3̄ ∶ 퐺ℚ,푆 ⟶ Aut(퐸[3]) ≃ GL2(픽3), ℓ (50) with ℓ-adic coefficients, one may consider the resulting 휚퐸,3 ∶ 퐺ℚ,푆 ⟶ GL2(ℤ3). mod ℓ representation The theorem of Langlands and Tunnell about the modular-

(47) 휚̄ ∶ 퐺ℚ,푆 ⟶ GL푛(픽ℓ). ity of the general quartic equation leads to the conclusion that 휚̄ is modular. This rests on the happy circumstance The passage from 휚 to 휚̄ amounts to replacing the 퐸,3 푟 that quantities 푁푝푟 (휚) ∈ ℤℓ as 푝 ranges over all the prime powers with their (51) GL2(픽3)/⟨±1⟩ ≃ 푆4, mod ℓ reduction. and hence that 퐸[3] has essentially the same symmetry Such a passage would Since then, group as the general quartic equation! The isomorphism seem rather contrived in (51) can be realised by considering the action of GL2(픽3) for the sequences “modularity lifting on the set {0, 1, 2, ∞} of points on the over 푁 푟 (풳)—why study 푝 픽3. the solution counts of theorems” have If 퐸 is semistable, Wiles is able to check that both a diophantine equa- 휚퐸,3 and 휚퐸,3̄ satisfy the conditions necessary to apply tion over different proliferated, and the Modularity Lifting Theorem, at least when 휚퐸,3̄ is finite fields, taken irreducible. It then follows that 휚퐸,3 is modular, and modulo ℓ?—if one their study, in ever therefore so is 퐸, since 휁(퐸; 푠) and 휁(휚퐸,3; 푠) are the same. did not know a pri- more general and Note the key role played by the result of Langlands– ori that these counts Tunnell in the above strategy. It is a dramatic illustration arise from ℓ-adic Ga- delicate settings, fo the unity and historical continuity of mathematics that lois representations the solution in radicals of the general quartic equation, with coefficients in ℤℓ. has spawned an one of the great feats of the algebraists of the Italian There is a correspond- industry. Renaissance, is precisely what allowed Langlands, Tunnell, ing notion of what it and Wiles to prove their modularity results more than means for 휚̄ to be mod- five centuries later. ular, namely, that the Having established the modularity of all semistable el- data of 푁 푟 (휚)̄ agrees, very loosely speaking, with the 푝 liptic curves 퐸 for which 휚퐸,3̄ is irreducible, Wiles disposes mod ℓ reduction of similar data arising from an automor- of the others by applying his lifting theorem to the prime phic representation. We can now state Wiles’s celebrated ℓ = 5 instead of ℓ = 3. The Galois representation 휚퐸,5̄ is modularity lifting theorem, which lies at the heart of his always irreducible in this setting, because no elliptic curve strategy: over ℚ can have a rational subgroup of order 15. Nonethe- Wiles’s Modularity Lifting Theorem. Let less, the approach of exploiting ℓ = 5 seems hopeless at first glance, because the Galois representation 퐸[5] is (48) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℤℓ) not known to be modular a priori, for much the same be an irreducible geometric Galois representation satisfy- reason that the general quintic equation cannot be solved ing a few technical conditions (involving, for the most part, by radicals. (Indeed, the symmetry group SL2(픽5) is a ̄ double cover of the alternating group 퐴 on 5 letters and the restriction of 휚 to the subgroup 퐺ℚℓ = Gal(ℚℓ/ℚℓ) of 5 퐺ℚ,푆). If 휚̄ is modular and irreducible, then so is 휚. thus is closely related to the symmetry group underlying the general quintic.) To establish the modularity of 퐸[5], This stunning result was completely new at the time: Wiles constructs an auxiliary semistable elliptic curve 퐸′ nothing remotely like it had ever been proved before! Since satisfying then, “modularity lifting theorems” have proliferated, and their study, in ever more general and delicate settings, has (52) 휚퐸̄ ′,5 = 휚퐸,5̄ , 휚퐸̄ ′,3 is irreducible. spawned an industry and led to a plethora of fundamental It then follows from the argument in the previous para- advances in the Langlands program. graph that 퐸′ is modular, hence that 퐸′[5] = 퐸[5] is Let us first explain how Wiles himself parlays his modular as well, putting 퐸 within striking range of the original modularity lifting theorem into a proof of the modularity lifting theorem with ℓ = 5. This lovely epi- Shimura–Taniyama-Weil conjecture for semistable elliptic logue of Wiles’s proof, which came to be known as the curves. Given such an elliptic curve 퐸, consider the groups “3-5 switch,” may have been viewed as an expedient trick (49) at the time. But since then the prime switching argument 푛 푛 푛 퐸[3 ] ∶= {푃 ∈ 퐸(ℚ)̄ ∶ 3 푃 = 0} , 푇3(퐸) ∶= lim 퐸[3 ], ← has become firmly embedded in the subject, and many

March 2017 Notices of the AMS 215 variants of it have been exploited to spectacular effect in • The Sato–Tate conjecture concerning the distribution deriving new modularity results. of the numbers 푁푝(퐸) for an elliptic curve 퐸 as the Wiles’s modularity prime 푝 varies, whose proof was known to follow lifting theorem reveals from the modularity of all the symmetric power The modularity of that “modularity is Galois representations attached to 퐸, was proved in contagious” and can large part by Laurent Clozel, Michael Harris, Nick elliptic curves was often be passed on to Shepherd-Barron, and Richard Taylor around 2006. an ℓ-adic Galois rep- • One can also make sense of what it should mean only the first in a resentation from its for diophantine equations over more general number mod ℓ reduction. It fields to be modular. The modularity of elliptic curves series of is this simple prin- over all real quadratic fields has been proved very ciple that accounts recently by Nuno Freitas, Bao Le Hung, and Samir Siksek spectacular for the tremendous by combining the ever more general and powerful applications. impact that the Modu- modularity lifting theorems currently available with a larity Lifting Theorem, careful diophantine study of the elliptic curves which and the many variants could a priori fall outside the scope of these lifting proved since then, continue to have on the subject. Indeed, theorems. the modularity of elliptic curves was only the first in a • Among the spectacular recent developments building series of spectacular applications of the ideas introduced on Wiles’s ideas is the proof, by Laurent Clozel and by Wiles, and since 1994 the subject has witnessed a Jack Thorne, of the modularity of certain symmetric real golden age, in which open problems that previously powers of the Galois representations attached to seemed completely out of reach have succumbed one by holomorphic modular forms, which is described in one. Thorne’s contribution to this volume. Among these developments, let us mention: These results are just a sampling of the transformative • The two-dimensional Artin conjecture, first formu- impact of modularity lifting theorems. The Langlands pro- lated in 1923, concerns the modularity of all odd, gram remains a lively area, with many alluring mysteries two-dimensional Galois representations yet to be explored. It is hard to predict where the next breakthroughs will come, but surely they will continue to (53) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℂ). capitalise on the rich legacy of Andrew Wiles’s marvelous The image of such a 휚 modulo the scalar matrices proof. is isomorphic either to a dihedral group, to 퐴4, to 푆 , or to 퐴 . Thanks to the earlier work of Hecke, 4 5 References Langlands, and Tunnell, only the case of projective [Da] H. Darmon, A proof of the full Shimura-Taniyama-Weil con- image 퐴 remained to be disposed of. Many new cases 5 jecture is announced, Notices of the AMS 46 (1999), no. 11, of the two-dimensional Artin conjecture were proved 1397–1401. MR1723249 in this setting by Kevin Buzzard, Mark Dickinson, Nick [Se] J-P. Serre, Lectures on 푁푋(푝), Chapman & Hall/CRC Re- Shepherd-Barron, and Richard Taylor around 2003 us- search Notes in Mathematics, 11, CRC Press, Boca Raton, ing the modularity of all mod 5 Galois representations FL. MR2920749 arising from elliptic curves as a starting point. • Serre’s conjecture, which was formulated in 1987, Photo Credits asserts the modularity of all odd, two-dimensional Photo of Wiles giving his first lecture is by C. J. Galois representations Mozzochi, courtesy of the Simons Foundation.

(54) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(픽푝푟 ), Photo of the conference in honor of Karl Rubin’s sixtieth birthday is courtesy of Kartik Prasanna. with coefficients in a finite field. This result was proved by Chandrasekhar Khare and Jean-Pierre Wintenberger in 2008 by a glorious extension of the “3-5 switching technique” in which essentially all the primes are used. (See Khare’s report in this volume.) This result also implies the two-dimensional Artin conjecture in the general case. • The two-dimensional Fontaine–Mazur conjecture con- cerning the modularity of odd, two-dimensional 푝-adic Galois representations

(55) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℚ̄ 푝) satisfying certain technical conditions with respect to their restrictions to the Galois group of ℚ푝. This theorem was proved in many cases as a consequence of work of Pierre Colmez, Matthew Emerton, and Mark Kisin.

216 Notices of the AMS Volume 64, Number 3