THE GRADUATE STUDENT SECTION

WHAT IS…

the Langlands Program?Solomon Friedberg

Communicated by Cesar E. Silva

The Langlands Program envisions deep links betweenarithmetic and analysis, and uses constructions in arith-metic to predict maps between spaces of functions ondifferent groups. The conjectures of the LanglandsProgram have shaped research in number theory, rep-resentation theory, and other areas for many years, butthey are very deep, and much still remains to be done.

For arithmetic, if 𝐾 is a finite Galois extension ofthe rationals we have the Galois group Gal(𝐾/ℚ) of allring automorphisms of 𝐾 fixing ℚ. Frobenius suggestedstudying groups by embedding them into groups ofmatrices, so fix a homomorphism 𝜌∶ Gal(𝐾/ℚ) → 𝐺𝐿(𝑉)where 𝑉 is a finite dimensional complex vector space. Foralmost all primes 𝑝 one can define a certain conjugacyclass Frob𝑝 in the Galois group. Artin introduced afunction in a complex variable 𝑠 built out of the valuesof the characteristic polynomials for 𝜌(Frob𝑝): 𝐿(𝑠,𝜌) =∏′

𝑝 det(I𝑉 −𝜌(Frob𝑝)𝑝−𝑠)−1. Here the product is an Eulerproduct, that is a product over the primes 𝑝; it convergesfor ℜ(𝑠) > 1. Also, there is a specific adjustment at afinite number of primes (indicated by the apostrophein the product above) which we suppress. For example,if 𝑉 is one dimensional and 𝜌 is trivial, then 𝐿(𝑠,𝜌) isexactly the Riemann zeta function. Artin made the deepconjecture that these “Artin 𝐿-functions” (which Brauershowed have meromorphic continuation to 𝑠 ∈ ℂ) areentire if 𝜌 is irreducible and nontrivial.

On the analysis side, let 𝐺 be a (nice) algebraic groupsuch as 𝐺𝐿𝑛. Then one can study the space 𝐿2(Γ\𝐺(ℝ))consisting of complex-valued functions on 𝐺(ℝ) that areinvariant under a large discrete subgroup Γ (such as afinite index subgroup of 𝐺𝐿𝑛(ℤ)), transform under the

Solomon Friedberg is James P. McIntyre professor of mathematicsat Boston College. His email address is solomon.friedberg@bc.edu.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1686

center by a character, and that are square-integrable withrespect to a natural group-invariant measure. The group𝐺(ℝ) acts by the right regular representation, and itturns out that to each invariant subspace 𝜋 (with someconditions) one can once again attach an Euler product𝐿(𝑠,𝜋) that converges for ℜ(𝑠) sufficiently large, calledthe standard 𝐿-function of 𝜋. The space of functions𝜋 is called an automorphic representation. (One mayalso consider certain other functions on Γ\𝐺(ℝ) andcertain quotient spaces.) The subject of automorphicforms studies the functions in 𝜋, and shows that formost 𝜋 the “automorphic” L-series 𝐿(𝑠,𝜋) is entire. Afirst example is 𝐺 = 𝐺𝐿1, and there one recovers (fromthe adelic version) the theory of Dirichlet 𝐿-functionsattached to a Dirichlet character 𝜒 ∶ (ℤ/𝑁ℤ)× → ℂ×, andused by Dirichlet to prove his primes in progressionstheorem.

Langlands’s first insight is that Artin’s conjectureshould be true because Galois representations are con-nected to automorphic representations. More precisely,he conjectures that each Artin 𝐿-function 𝐿(𝑠,𝜌) shouldin fact be an automorphic 𝐿-function 𝐿(𝑠,𝜋), with 𝜋 on𝐺𝐿dim𝑉. If dim𝑉 = 1 this assertion states that the Artin𝐿-function is in fact a Dirichlet 𝐿-function; this is Artin’sfamous reciprocity law. Incidentally, the converse to thisconjecture is false—there are far more analytic objectsthan algebraic ones.

This is already remarkable. But Langlands suggestsmuch more. There are natural (and easy) algebraic con-structions on the arithmetic side. Langlands conjecturesthat there should be matching (but, it seems, not easy)constructions on the analytic side.

For example, suppose 𝑉 and 𝑊 are two finite dimen-sional vector spaces and 𝜎 ∶ 𝐺𝐿(𝑉) → 𝐺𝐿(𝑊) is a grouphomomorphism. (Concretely, 𝑊 could be the symmetric𝑛th power Sym𝑛(𝑉) for some fixed 𝑛 and 𝜎 the naturalmap.) Then there is a map on the Galois side given bycomposition: 𝜌 ↦ 𝜎∘ 𝜌. If each of these Galois represen-tations corresponds to an automorphic representation,

June/July 2018 Notices of the AMS 663

THE GRADUATE STUDENT SECTIONGal(𝐾/ℚ)

𝜎∘𝜌

&&LLLLL

LLLLL

𝜌 // 𝐺𝐿(𝑉)

𝜎��

///o/o/o/o 𝐿(𝑠,𝜌) a special case of 𝐿(𝑠,𝜋) 𝜋 on 𝐺𝐿dim𝑉oo o/ o/ o/

Functoriality��

𝐺𝐿(𝑊) ///o/o/o 𝐿(𝑠,𝜎 ∘ 𝜌) a special case of 𝐿(𝑠,Π) Π on 𝐺𝐿dim𝑊oo o/ o/ o/

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

Gal(𝐾/ℚ)𝜎∘𝜌

%%KKKKK

KKKKKK

𝜌 // 𝐻(ℂ)� _𝜎��

///o/o/o 𝐿(𝑠,𝜌) a special case of 𝐿(𝑠,𝜋) 𝜋 on dual group 𝐺oo o/ o/ o/

EndoscopicTransfer

��𝐺𝐿(𝑉) ///o/o/o 𝐿(𝑠,𝜎∘𝜌)

=𝐿(𝑠,𝜌) a special case of 𝐿(𝑠,Π)=𝐿(𝑠,𝜋) Π on 𝐺𝐿dim𝑉oo o/ o/ o/ o/ o/

Figure 2. Endoscopic transfer: automorphic representations on different groups with the same 𝐿-function.

then there should be a map taking those spaces of func-tions attached to 𝐺𝐿dim𝑉 to certain spaces of functionsattached to 𝐺𝐿dim𝑊. The Langlands Functoriality Conjec-ture asserts that there should be a full matching mapon the analytic side, taking general automorphic repre-sentations 𝜋 on quotients of 𝐺𝐿dim𝑉(ℝ) to automorphicrepresentations Π on quotients of 𝐺𝐿dim𝑊(ℝ), such thatthe 𝐿-functions correspond. (More about this momentar-ily.) See Figure 1. There is no simple reason why theanalysis on these two groups, each modulo a discretesubgroup, should be related; indeed, even if one can finda natural way to construct a function on 𝐺𝐿dim𝑊(ℝ) fromone on 𝐺𝐿dim𝑉(ℝ), it is quite difficult to make one that isinvariant under a large discrete subgroup of 𝐺𝐿dim𝑊(ℝ)and square-integrable on the quotient. The existence ofthis map in general would have important consequencesincluding the generalized Ramanujan Conjecture. It isknown in only a few cases.

The correspondence of 𝐿-functions is a key feature ofthe Langlands Functoriality Conjecture. For the conjec-tural map 𝜋 ↦ Π described above, this means that if (forℜ(𝑠) sufficiently large)

𝐿(𝑠,𝜋) = ∏′

𝑝det(Idim𝑉 −𝐴𝑝𝑝−𝑠)−1

where 𝐴𝑝 is an invertible diagonal matrix for all butfinitely many primes 𝑝, then the new product

𝐿(𝑠,𝜋,𝜎) ∶= ∏′

𝑝det(Idim𝑊 −𝜎(𝐴𝑝)𝑝−𝑠)−1

matches 𝐿(𝑠,Π) for almost all primes 𝑝. In fact, similarlyto the Riemann zeta function, all automorphic 𝐿-functionsshould have analytic continuation to ℂ and satisfy func-tional equations under 𝑠 ↦ 1− 𝑠. We know this propertyfor 𝐿(𝑠,𝜋) but not for 𝐿(𝑠,𝜋,𝜎) in general (let alone that𝐿(𝑠,𝜋,𝜎) matches the standard 𝐿-function for an auto-morphic representation Π on 𝐺𝐿dim𝑊). The continuationof 𝐿(𝑠,𝜋,𝜎) to ℂ for a given symmetric power 𝜎 alreadycarries important analytic information.

Just as Langlands predicts a map of spaces of func-tions that corresponds to the homomorphism 𝜎, he

also predicts maps of automorphic representations cor-responding to other algebraic operations: a map that is acounterpart of the algebraic operation of tensor product,and maps corresponding to induction and restriction ofGalois representations. These lastmaps require a tower offield extensions and base fields other than ℚ, a topic thatis best addressed by passing to the adeles. The automor-phic induction and restriction maps for the general lineargroups were constructed for cyclic extensions by Arthurand Clozel using the trace formula, following earlier workfor 𝐺𝐿2 by Langlands and others.

So far we have only made use of general linear groups.Langlands also addresses how other algebraic groupsenter the story. Suppose that the image of the Galoisrepresentation 𝜌 is contained in a subgroup 𝐻(ℂ) of𝐺𝐿(𝑉), which is the complex points of a (nice) algebraicgroup 𝐻. Then Langlands conjectures that there shouldbe an automorphic representation 𝜋 on a group 𝐺, thedual group of𝐻 (the definition involves Lie theoretic data),whose 𝐿-functionmatches that of𝜌. For example, supposethe image of 𝜌 stabilizes a nondegenerate symplecticform on 𝑉, so after a choice of basis, 𝜌 maps intothe symplectic group 𝑆𝑝𝑛(ℂ) ⊂ 𝐺𝐿𝑛(ℂ) for some evennumber 𝑛. Then Langlands conjectures that 𝜌 shouldcorrespond (in the sense of matching 𝐿-functions) to anautomorphic representation 𝜋 on the special orthogonalgroup 𝑆𝑂𝑛+1.

But wait amoment! If we simply forget that𝜌maps intoa subgroup𝐻(ℂ) of𝐺𝐿(𝑉), then𝜌 should also correspondto an automorphic representationΠ of𝐺𝐿dim𝑉. Langlandsthen suggests that every automorphic representation 𝜋of 𝐺 should correspond to an automorphic representa-tion Π of 𝐺𝐿dim𝑉, with 𝐿(𝑠,𝜋) = 𝐿(𝑠,Π). See Figure 2.To summarize, given an automorphic representation ona symplectic or orthogonal group, there should be acorresponding automorphic representation on a generallinear group with the same 𝐿-function, correspondingto the natural inclusion on the dual group side. Suchmaps are called endoscopic liftings (they ‘see inside’ thegeneral linear group). They were established for certainclasses of automorphic representations by Cogdell, Kim,

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THE GRADUATE STUDENT SECTION

Figure 3. In the 1960s and 1970s Robert Langlandsproposed a program relating arithmetic and certainspaces of functions on groups called automorphicrepresentations.

Piatetski-Shapiro and Shahidi, and in full generality usingthe trace formula by Arthur.

More generally, whenever there is a homomorphismof dual groups the Langlands Functoriality Conjectureasserts that there should be a corresponding map ofautomorphic representations. That is, spaces of functionson thequotients ofdifferent real (ormoregenerally, adelic)groups by discrete subgroups should be related. A furtherextension of functoriality replaces the dual group by the𝐿-group, which involves the Galois group, so that theautomorphicity of Artin 𝐿-functions is part of the samefunctoriality conjecture. Langlands is describing this inFigure 3.

The endoscopic lifting established by Arthur is asignificant step in the Langlands Program. However, mostof the program, including most cases of establishingfunctorial liftings of automorphic representations andmost cases of the matching of Artin and automorphic 𝐿-functions, remains unproved. There is also a conjecturedvast generalization of this matching, that every motivic 𝐿-function over a number field is an automorphic 𝐿-function(this includes Wiles’s Theorem as one case), that is mostlyunproved.

The Langlands Program also includes local questions,which allow one to say more about the 𝐿-functions at thefinite number of primes 𝑝 that were not discussed above(and where much progress has recently been announcedby Scholze), and a versionwhere a number field is replacedby the function field attached to a curve over a finite field,resolved for the general linear groups by L. Lafforgue.Another direction is the 𝑝-adic Langlands Program whereone considers 𝑝-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 Riemannsurfaces. This substitution leads to the geometric Lang-lands Program, with connections to physics. The richnessof Langlands’s fundamental idea that simple naturalmaps

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

EDITOR’S NOTE. Robert Langlands in March wasnamed winner of the 2018 Abel Prize for his pro-gram. More on Langlands is planned for the 2019Notices. Meanwhile see the feature on James G. Arthurand his work on the Langlands Program in this issue.

Image CreditsFigures 1 and 2 by author.Figure 3 photo by Jeff Mozzochi, courtesy of the Simons Founda-

tion.Author photo by Lee Pellegrini (Boston College).

References[1] J. Arthur, The principle of functoriality, Bull. Amer. Math.

Soc. (N.S.) 40 (2003), no. 1, 39–53. MR1943132[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.MR1990371

[3] E. Frenkel, Elementary introduction to the Langlandsprogram, four video lectures (MSRI, 2015), available athttps://www.msri.org/workshops/804/schedules or onYouTube.

[4] S. Gelbart, An elementary introduction to the Langlands pro-gram. Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219.MR733692

[5] The Work of Robert Langlands, papers, letters and commen-tary available from the Institute for Advanced Study website,publications.ias.edu/rpl.

ABOUT THE AUTHOR

In addition to his work in numbertheory and representation theoryand involvement in K–12math edu-cation, Solomon Friedberg enjoysoutdoor activities, the arts, andforeign travel.

Solomon Friedberg

June/July 2018 Notices of the AMS 665