Robert Langlands Wins the 2018 Abel Prize

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Robert Langlands Wins tuted. Niels Henrik Abel is the best-known among the famous Norwegian triad – others the 2018 Abel Prize being Sophus Lie and Ludwig Sylow. Dur- ing his short life span of August 1802–April B Sury 1829, Abel’s outstanding contributions were yet to be recognized. In impoverished circum- The so-called Langlands program is a kind of stances, he tragically died of tuberculosis just grand unification in mathematics. In 1983, two days before a letter appointing him to a abriefing panel of the National Academy of professorship at the university in Berlin was Sciences, USA, chaired by William Browder received. To show how ubiquitous the name ffi presented to the O ce of Science and Tech- of Abel is, we observe that it appears in un- nology of the President of the United States a dergraduate texts as the adjective ‘abelian’ to summary of progress in theoretical mathemat- signify an operation which is commutative. ics; among other things, it mentioned : In 1899, Lie proposed an Abel Prize to be “The unifying role of group symmetry in ge- instituted to commemorate the centenary of ometry, so penetratingly expounded by Fe- Abel’s birth in 1902. However, the initial en- lix Klein in his 1872 Erlanger Program, has thusiastic response died off when Lie passed led to a century of progress. A worthy suc- away. Thereafter, it took another century for cessor to the Erlanger Program seems to be the Prize to be instituted. The first Abel Prize Langlands’s program to use infinite dimen- was awarded in 2003 to Jean-Pierre Serre. The sional representations of Lie groups to illumi- 2015 Prize was the only one shared between nate number theory. That the possible number two people – John Nash and Louis Nirenberg. fields of degree n are restricted in nature by It may be recalled that while returning home the irreducible infinite dimensional represen- after the Prize ceremony, Nash and his wife tations of GL(n) was the visionary conjecture were shockingly killed in a road accident. of R P Langlands. His far-reaching conjec- How did the Langlands Program originate? tures present tantalizing problems whose so- In January 1967, Langlands wrote a letter to lution will lead us to a better understanding Andre´ Weil, outlining some of his new math- of representation theory, number theory and ematical insights with the prelude: “If you algebraic geometry. Impressive progress has are willing to read it as pure speculation I already been made, but very much more lies would appreciate that; if not I am sure you ahead.” have a waste basket handy.” Prior to this, For this visionary program “connecting repre- in the beginning of January, Langlands and sentation theory to number theory”, the Abel Weil met incidentally in a corridor of the In- Prize for Mathematics 2018 has been awarded stitute for Defense Analysis in Princeton, both to Robert P Langlands from the Institute for having arrived early for a lecture by Shiing- Advanced Study Princeton, USA. One may Shen Chern. Not knowing quite how to start a say that this was overdue. conversation, Langlands began to describe his reflections of the connections between num- Here is how the Abel Prize came to be insti- ber theory and automorphic forms. Weil, us-

RESONANCE | May 2018 613 RESEARCH NEWS ing “a well-known stratagem to escape po- quotient being abelian. This shows that study- litely from importunate individuals” (in Lang- ing number theory is facilitated by the study lands’s words), suggested that the young col- of Galois groups (a name given to the above- league could send him a letter describing his mentioned group of automorphisms). thoughts. The result was that famous 17-page About a 100 years back, Artin, Hilbert, letter. Hasse et al., developed methods to generalize Three years later, Langlands made some bold Gauss’s quadratic reciprocity law in the fol- conjectures which served as bridges between lowing manner. Let us lead to this gently. If number theory and harmonic analysis (two ap- d is a square-free integer, quadratic reciprocity parently unrelated disciplines). Many of these law helps us decide whether d is a square mod- conjectures are yet to be proven but the ef- ulo any prime p. Namely, for simplicity, con- forts to prove them have driven the course of sider a prime q p (both odd). Then, q is a modern mathematical research directly or in- square modulo p if and only if p is a square directly. The 2002 and 2010 Fields Medals modulo q (unless both p, q are 3 modulo 4 were awarded for work on the Langlands con- in which case the opposite implication holds). jectures. For instance, the fact that 3 is a square modulo 71 can be deduced immediately (although Before embarking on a description of the finding the square roots 28 and 43 are not that actors involved in the Langlands program, quick) because 71 modulo 3 is 2, which is we mention in passing that the solution of not a square modulo 3. The quadratic reci- the Shimura-Taniyama-Weil conjecture (and procity law can be viewed in the following√ therefore, of Fermat’s last theorem) is a fall- way too. The realm of numbers a + b d as out of the Langlands program. A very ba- a, b vary over rational numbers, forms a ‘field’ sic aim in number theory is to solve polyno- (where we can add and subtract and multiply mial equations over the integers. The theory and divide within itself similarly to what we of Galois transfers such problems into prob- do among rational√ or real or complex num- lems in the realm of group theory. In fact, bers) denoted Q[ d]. The set of those ele- Galois proved that if f is a polynomial with ments which satisfy a monic integer polyno- integer coefficients, the roots are expressible mial is the ‘ring of integers’ O√d of this field. in terms of the coefficients via algebraic op- Here O is an analogue for Q[ d] of what Z erations and taking square, cube and higher d is for the field Q – one can add and subtract roots if and only if the following property and also multiply but cannot always divide in holds good. Look at the finite group of those a ring! However, it may lack a unique way permutations of the roots which act as ‘auto- to factor elements into ‘irreducible’ elements morphisms’ on the smallest realm containing akin to the prime factorization of integers.√ For the roots of f ; then this group must have an instance,√ if d = −5, the numbers 1 + 5i and algebraic property - now called solvability – 1 − 5i are irreducible in this ring and their which is an approximation to being commuta- product is also 2×3 which is another factoriza- tive. More precisely, this permutation group tion. However, the uniqueness of factorization must admit of a filtration with each successive is recovered in such number rings when one

614 RESONANCE | May 2018 RESEARCH NEWS considers ‘ideals’ instead of single elements stance, if K is the field obtained by attaching – roughly, one considers the whole package the n-th roots of 1 for some n, the Frobenius of multiples in that ring of an element rather automorphism corresponding to a prime p is than just an individual element. The notion trivial if and only if p ≡ 1 modulo n.IfK is of prime ideals generalizes that of prime num- a general abelian extension of Q, the Artin’s bers and uniqueness of factorization of ideals reciprocity law asserts that for any character into prime ideals is valid. A prime number σ : Gal(K/Q) → C∗, there exists a positive p gives rise to an ideal pOd consisting of all integer (the smallest is called the conductor of multiples of p in such a ring Od. How this σ) Nσ and a homomorphism χσ (also called ∗ ideal factors into prime ideals is governed by a Dirichlet character mod Nσ) from (Z/NσZ) the question of whether d is a square modulo to C∗ such that: p or not. Indeed, it turns out that d is a square σ = χ , modulo p (that is, x2 − d = 0 has two solu- (Frobp) σ(p) tions modulo p) if and only if pO factorizes d for almost all the primes p (leaving out the√ fi- into two different prime ideals. For instance, nite set of ramified primes; in the case Q[ d], if d = −1, then p is congruent to 1 modulo 4 if these are the primes dividing d). and only if p is expressible as x2 + y2 – this is the famous ‘two-squares theorem’ of Fermat. √ A prime number p is said to ‘split completely’ By automorphisms of a field like Q[ d], one in a number field K if the integer polyno- means bijective transformations which respect mial one of whose roots generates K factor- both its addition and its multiplication. For izes modulo p into linear factors. The set instance, complex conjugation gives an au- S (K) of primes splitting in a number field K tomorphism of Q[i]. In terms of the Ga- determines it completely. So, the real chal- lois group G (the group of all field automor- √ lenge is to characterize the sets S (K) purely in phisms of Q[ d]), one has a well-defined terms of Q. Any such characterization can be group homomorphism from the two element called a ‘reciprocity law’. Abelian class field group {1, 3} to G given by sending a prime theory accomplishes this in case of abelian p modulo 4 (for primes p not dividing d)to extensions by coming up with a positive in- d the quadratic symbol p whose values are 1 teger N such that the elements of S (K)are or −1 according as to whether d is a square precisely the primes in certain residue classes or not modulo p. The fact that this depends modulo N. As an instance, a statement in only on the residue class of p modulo 4 is a the previous paragraph implies that a prime crucial aspect of Artin’s reciprocity law. The splits completely in the field K obtained by above mapping is called√ the Artin map more attaching the n-th roots of unity if and only generally when Q[ d] is replaced by a finite if p ≡ 1 modulo n. The success of abelian extension field K of Q whose Galois group is class field theory behoves us to look for reci- abelian; the automorphism in the Galois group procity laws for nonabelian extensions K of which corresponds to a prime p is called the Q. Note that p splits completely if and only if Frobenius at p denoted as Frobp. For in- the Frobenius conjugacy class Frobp is trivial; hence, the principal aim is to describe the fam-

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ily {Frobp}p in terms of Q. In the nonabelian ture. The analytic properties of the L-function case, each prime (barring finitely many) gives of an automorphic representation like its an- a conjugacy class in the nonabelian Galois alytic continuation and functional equation group Gal(K/Q). The famous density theorem were proved by Langlands when n = 2; for of Chebotarev shows that a particular conju- general n he obtained their meromorphic con- gacy class C in the Galois group Gal(K/Q)is tinuation using the theory of Eisenstein series. represented by a proportion of primes equal Remarkably, Langlands’s reciprocity conjec- to |C|/|Gal(K/Q)|. Artin associated to an n- ture subsumes the Artin reciprocity law and dimensional representation ρ : Gal(K/Q) → many other classical themes. The so-called GLn(C), an L-function which is the product: Sato-Tate conjecture (now proved), Selberg’s (1/4)-eigenvalue conjecture, and generalized −s −1 L(s,ρ) = det(In − ρ(Frobp)p ) . Ramanujan’s conjecture would follow from p Langlands’s reciprocity conjecture. Note that the factors are the characteristic polynomials of Frobp for various primes p. The other conjectures – specifically, Lang- Note that when the Galois extension is abelian lands’s Functoriality conjecture – require the (and n = 1), the product defining the L- notion of L-groups of a reductive algebraic function coincides with the Dirichlet series group G; this is a complex Lie group LG. χ(n) n ns corresponding to a Dirichlet character. ‘Nice’ representations of the adelic group of In general, Artin proved analytic properties of G give a family of conjugacy classes {Cv}v the L-function L(s,ρ). (for various primes v) in the complex Lie group LG because LG is constructed using the Therefore, the L-function L(s,ρ) coincides root system of G.ForG = GL ,theL- with the L-function of a certain Dirichlet char- n group is GL (C). Then, varying the homo- acter when K is abelian. However, there was n morphisms ρ : GL (C) → GL (C), yields no known object corresponding to Dirichlet n n a family of L-functions for GL . Another characters for a general nonabelian Galois ex- n conjecture of Langlands considers an auto- tension. Roughly, Langlands produced such morphic representation π of G and a finite- a corresponding object. He defined the no- dimensional representation of its L-group LG. tion of an automorphic representation of GL ; n It asserts that the Euler product L(s, Π, r) ini- this is a representation of the ‘adelic group’ tially defined in a right half-plane of s, can GL (AQ), the group obtained by taking the n be meromorphically continued to all of C ‘restricted’ product of all the groups GL (Q ) n p and admits a functional equation which re- and GL (R). Langlands boldly conjectured n lates L(s, Π, r) and L(1−s, Π˜ , r). Langlands that for any n-dimensional representation ρ of conjectured the existence of a locally com- a Galois group Gal(K/Q), there corresponds pact group mapping onto the Galois group – an automorphic representation πρ of GL such n this is now called an ‘automorphic Langlands that L(s,ρ) = L(s,πρ). The definition of the group’ – whose continuous, n-dimensional right hand side is quite involved and we avoid representations parametrize automorphic rep- giving it precisely here. The above conjecture resentations of GL(n). Its existence is as yet is known as Langlands’s Reciprocity conjec-

616 RESONANCE | May 2018 RESEARCH NEWS not known! Finally, Langlands’s functorial- an undergraduate degree in 1957, an MSc in ity conjecture(which implies the above con- 1958, and from Yale University with a PhD jectures – the reciprocity conjecture can be in 1960. He has held faculty positions at regarded as the special case when one of the Princeton University and Yale University, and groups is trivial) asserts that a homomorphism is currently a Professor at the Institute for between L-groups of two reductive groups G Advanced Study in Princeton, USA. Before and G leads to a correspondence between au- the Abel Prize, Langlands has won several tomorphic representations of G and G.The recognitions and accolades for his deep con- function ﬁeld analogue of the functoriality tributions to mathematics. Some of these are conjecture has been proved for GLn by L Laf- ‘The Shaw Prize’, ‘The Nemmers Prize’, ‘The forgue. The proof of semistability of ellip- Wolf Prize’ (jointly with Andrew Wiles), ‘The tic curves over Q by Andrew Wiles can be Steele Prize’, and ‘The Cole Prize’. regarded as an instance of Langlands’s Reci- procity Conjecture in the n = 2 case. The Langlands’ program covers all bases, local version of the Langlands program leads brushes aside class ﬁelds, Hecke theory as to the local Langlands conjectures which have special cases. been proved for GL in any characteristic. The n Its consequences – far too many to list, importance of Langlands’s conjectures is not and some of its tools – not even known to only in explaining and unifying old results and exist; solving old problems but also in giving rise to a quandary that one sometimes faces! other interesting problems for study. Robert Langlands was born in New Westmin- ster, British Columbia, in 1936. He graduated B Sury, Indian Statistical Institute, Bangalore from the University of British Columbia with Email: [email protected]

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