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WHATIS… the Langlands Program? Solomon Friedberg Communicated by Cesar E. Silva

The Langlands Program envisions deep links between center by a character, and that are square-integrable with arithmetic and analysis, and uses constructions in arith- respect to a natural group-invariant measure. The group metic to predict maps between spaces of functions on 퐺(ℝ) acts by the right regular representation, and it different groups. The of the Langlands turns out that to each invariant subspace 휋 (with some Program have shaped research in , rep- conditions) one can once again attach an Euler product resentation theory, and other areas for many years, but 퐿(푠, 휋) that converges for ℜ(푠) sufficiently large, called they are very deep, and much still remains to be done. the standard 퐿-function of 휋. The space of functions For arithmetic, if 퐾 is a finite of 휋 is called an automorphic representation. (One may the rationals we have the Gal(퐾/ℚ) of all also consider certain other functions on Γ\퐺(ℝ) and ring automorphisms of 퐾 fixing ℚ. Frobenius suggested certain quotient spaces.) The subject of automorphic studying groups by embedding them into groups of forms studies the functions in 휋, and shows that for matrices, so fix a homomorphism 휌∶ Gal(퐾/ℚ) → 퐺퐿(푉) most 휋 the “automorphic” L-series 퐿(푠, 휋) is entire. A where 푉 is a finite dimensional complex vector space. For first example is 퐺 = 퐺퐿1, and there one recovers (from almost all primes 푝 one can define a certain conjugacy the adelic version) the theory of Dirichlet 퐿-functions × × class Frob푝 in the Galois group. Artin introduced a attached to a Dirichlet character 휒 ∶ (ℤ/푁ℤ) → ℂ , and function in a complex variable 푠 built out of the values used by Dirichlet to prove his primes in progressions of the characteristic polynomials for 휌(Frob푝): 퐿(푠, 휌) = theorem. ′ −푠 −1 ∏푝 det(I푉 − 휌(Frob푝)푝 ) . Here the product is an Euler Langlands’s first insight is that Artin’s product, that is a product over the primes 푝; it converges should be true because Galois representations are con- for ℜ(푠) > 1. Also, there is a specific adjustment at a nected to automorphic representations. More precisely, finite number of primes (indicated by the apostrophe he conjectures that each Artin 퐿-function 퐿(푠, 휌) should in the product above) which we suppress. For example, in fact be an automorphic 퐿-function 퐿(푠, 휋), with 휋 on if 푉 is one dimensional and 휌 is trivial, then 퐿(푠, 휌) is 퐺퐿dim 푉. If dim 푉 = 1 this assertion states that the Artin exactly the . Artin made the deep 퐿-function is in fact a Dirichlet 퐿-function; this is Artin’s conjecture that these “Artin 퐿-functions” (which Brauer famous . Incidentally, the converse to this showed have meromorphic continuation to 푠 ∈ ℂ) are conjecture is false—there are far more analytic objects entire if 휌 is irreducible and nontrivial. than algebraic ones. On the analysis side, let 퐺 be a (nice) This is already remarkable. But Langlands suggests 2 such as 퐺퐿푛. Then one can study the space 퐿 (Γ\퐺(ℝ)) much more. There are natural (and easy) algebraic con- consisting of complex-valued functions on 퐺(ℝ) that are structions on the arithmetic side. Langlands conjectures invariant under a large discrete subgroup Γ (such as a that there should be matching (but, it seems, not easy) finite index subgroup of 퐺퐿푛(ℤ)), transform under the constructions on the analytic side. For example, suppose 푉 and 푊 are two finite dimen- Solomon Friedberg is James P. McIntyre professor of sional vector spaces and 휎 ∶ 퐺퐿(푉) → 퐺퐿(푊) is a group at Boston College. His email address is solomon.friedberg@bc homomorphism. (Concretely, 푊 could be the symmetric .edu. 푛th power Sym푛(푉) for some fixed 푛 and 휎 the natural For permission to reprint this article, please contact: map.) Then there is a map on the Galois side given by [email protected]. composition: 휌 ↦ 휎 ∘ 휌. If each of these Galois represen- DOI: http://dx.doi.org/10.1090/noti1686 tations corresponds to an automorphic representation,

June/July 2018 Notices of the AMS 663 THEGRADUATESTUDENTSECTION

휌 / /o /o /o /o / o o/ o/ o/ Gal(퐾/ℚ)L 퐺퐿(푉) 퐿(푠, 휌) a special case of 퐿(푠, 휋) 휋 on 퐺퐿dim 푉 LLL L휎∘휌L LLL 휎 Functoriality L&   /o /o /o / o o/ o/ o/ 퐺퐿(푊) 퐿(푠, 휎 ∘ 휌) a special case of 퐿(푠, Π) Π on 퐺퐿dim 푊

Figure 1. Natural maps for Galois representations lead to conjectured functoriality maps.

휌 / /o /o /o / o o/ o/ o/ Gal(퐾/ℚ)KK 퐻(ℂ) _ 퐿(푠, 휌) a special case of 퐿(푠, 휋) 휋 on dual group 퐺 KKK K휎∘휌 Endoscopic KK 휎 KK Transfer K%   /o /o /o / 퐿(푠,휎∘휌) 퐿(푠,Π) o o/ o/ o/ o/ o/ 퐺퐿(푉) =퐿(푠,휌) a special case of =퐿(푠,휋) Π on 퐺퐿dim 푉

Figure 2. Endoscopic transfer: automorphic representations on different groups with the same 퐿-function. then there should be a map taking those spaces of func- also predicts maps of automorphic representations cor- tions attached to 퐺퐿dim 푉 to certain spaces of functions responding to other algebraic operations: a map that is a attached to 퐺퐿dim 푊. The Langlands Functoriality Conjec- counterpart of the algebraic operation of tensor product, ture asserts that there should be a full matching map and maps corresponding to induction and restriction of on the analytic side, taking general automorphic repre- Galois representations. These last maps require a tower of sentations 휋 on quotients of 퐺퐿dim 푉(ℝ) to automorphic field extensions and base fields other than ℚ, a topic that representations Π on quotients of 퐺퐿dim 푊(ℝ), such that is best addressed by passing to the adeles. The automor- the 퐿-functions correspond. (More about this momentar- phic induction and restriction maps for the general linear ily.) See Figure 1. There is no simple reason why the groups were constructed for cyclic extensions by Arthur analysis on these two groups, each modulo a discrete and Clozel using the trace formula, following earlier work subgroup, should be related; indeed, even if one can find for 퐺퐿2 by Langlands and others. So far we have only made use of general linear groups. a natural way to construct a function on 퐺퐿dim 푊(ℝ) from Langlands also addresses how other algebraic groups one on 퐺퐿dim 푉(ℝ), it is quite difficult to make one that is enter the story. Suppose that the image of the Galois invariant under a large discrete subgroup of 퐺퐿dim 푊(ℝ) and square-integrable on the quotient. The existence of representation 휌 is contained in a subgroup 퐻(ℂ) of this map in general would have important consequences 퐺퐿(푉), which is the complex points of a (nice) algebraic including the generalized Ramanujan Conjecture. It is group 퐻. Then Langlands conjectures that there should known in only a few cases. be an automorphic representation 휋 on a group 퐺, the The correspondence of 퐿-functions is a key feature of dual group of 퐻 (the definition involves Lie theoretic data), the Langlands Functoriality Conjecture. For the conjec- whose 퐿-function matches that of 휌. For example, suppose tural map 휋 ↦ Π described above, this means that if (for the image of 휌 stabilizes a nondegenerate symplectic form on 푉, so after a choice of basis, 휌 maps into ℜ(푠) sufficiently large) the symplectic group 푆푝푛(ℂ) ⊂ 퐺퐿푛(ℂ) for some even ′ −푠 −1 퐿(푠, 휋) = ∏ det(Idim 푉 − 퐴푝푝 ) number 푛. Then Langlands conjectures that 휌 should 푝 correspond (in the sense of matching 퐿-functions) to an where 퐴푝 is an invertible diagonal matrix for all but automorphic representation 휋 on the special orthogonal finitely many primes 푝, then the new product group 푆푂푛+1. ′ −푠 −1 But wait a moment! If we simply forget that 휌 maps into 퐿(푠, 휋, 휎) ∶= ∏ det(Idim 푊 − 휎(퐴푝)푝 ) a subgroup 퐻(ℂ) of 퐺퐿(푉), then 휌 should also correspond 푝 to an automorphic representation Π of 퐺퐿dim 푉. Langlands matches 퐿(푠, Π) for almost all primes 푝. In fact, similarly then suggests that every automorphic representation 휋 to the Riemann zeta function, all automorphic 퐿-functions of 퐺 should correspond to an automorphic representa- should have analytic continuation to ℂ and satisfy func- tion Π of 퐺퐿dim 푉, with 퐿(푠, 휋) = 퐿(푠, Π). See Figure 2. tional equations under 푠 ↦ 1 − 푠. We know this property To summarize, given an automorphic representation on for 퐿(푠, 휋) but not for 퐿(푠, 휋, 휎) in general (let alone that a symplectic or orthogonal group, there should be a 퐿(푠, 휋, 휎) matches the standard 퐿-function for an auto- corresponding automorphic representation on a general morphic representation Π on 퐺퐿dim 푊). The continuation linear group with the same 퐿-function, corresponding of 퐿(푠, 휋, 휎) to ℂ for a given symmetric power 휎 already to the natural inclusion on the dual group side. Such carries important analytic information. maps are called endoscopic liftings (they ‘see inside’ the Just as Langlands predicts a map of spaces of func- ). They were established for certain tions that corresponds to the homomorphism 휎, he classes of automorphic representations by Cogdell, Kim,

664 Notices of the AMS Volume 65, Number 6 THEGRADUATESTUDENTSECTION

for Galois representations are the shadows of maps for all automorphic representations continues to expand.

EDITOR’S NOTE. in March was named winner of the 2018 for his pro- gram. More on Langlands is planned for the 2019 Notices. Meanwhile see the feature on James G. Arthur and his work on the Langlands Program in this issue.

Image Credits Figures 1 and 2 by author. Figure 3 photo by Jeff Mozzochi, courtesy of the Simons Founda- tion. Author photo by Lee Pellegrini (Boston College). Figure 3. In the 1960s and 1970s Robert Langlands proposed a program relating arithmetic and certain References spaces of functions on groups called automorphic [1] J. Arthur, The principle of functoriality, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 1, 39–53. MR1943132 representations. [2] D. Bump, J. W. Cogdell, E. de Shalit, D. Gaitsgory, E. Kowalski, and S. S. Kudla, An introduction to the Lang- lands program, Birkhäuser Boston, Inc., Boston, MA, 2003. Piatetski-Shapiro and Shahidi, and in full generality using MR1990371 the trace formula by Arthur. [3] E. Frenkel, Elementary introduction to the Langlands More generally, whenever there is a homomorphism program, four video lectures (MSRI, 2015), available at of dual groups the Langlands Functoriality Conjecture https://www.msri.org/workshops/804/schedules or on asserts that there should be a corresponding map of YouTube. automorphic representations. That is, spaces of functions [4] S. Gelbart, An elementary introduction to the Langlands pro- gram. Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219. on the quotients of different real (or more generally, adelic) MR733692 groups by discrete subgroups should be related. A further [5] The Work of Robert Langlands, papers, letters and commen- extension of functoriality replaces the dual group by the tary available from the Institute for Advanced Study website, 퐿-group, which involves the Galois group, so that the publications.ias.edu/rpl. automorphicity of Artin 퐿-functions is part of the same functoriality conjecture. Langlands is describing this in Figure 3. The endoscopic lifting established by Arthur is a ABOUT THE AUTHOR significant step in the Langlands Program. However, most In addition to his work in number of the program, including most cases of establishing theory and functorial liftings of automorphic representations and and involvement in K–12 math edu- most cases of the matching of Artin and automorphic 퐿- cation, Solomon Friedberg enjoys functions, remains unproved. There is also a conjectured outdoor activities, the arts, and vast generalization of this matching, that every motivic 퐿- foreign travel. function over a number field is an automorphic 퐿-function (this includes Wiles’s Theorem as one case), that is mostly Solomon Friedberg unproved. The Langlands Program also includes local questions, which allow one to say more about the 퐿-functions at the finite number of primes 푝 that were not discussed above (and where much progress has recently been announced by Scholze), and a version where a number field is replaced by the function field attached to a curve over a finite field, resolved for the general linear groups by L. Lafforgue. Another direction is the 푝-adic Langlands Program where one considers 푝-adic representations. Langlands’s vision connects arithmetic and analysis, and begins from Galois groups. We also know that Galois- like groups may be attached to coverings of Riemann surfaces. This substitution leads to the geometric Lang- lands Program, with connections to physics. The richness of Langlands’s fundamental idea that simple natural maps

June/July 2018 Notices of the AMS 665