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Research News

The Nonabelian Reciprocity Law for Local Fields Jonathan Rogawski

This note reports on the Local Langlands Corre- sible to formulate a unified and general reciproc- spondence for GLn over a p-adic field, which was ity law until the notion of a reciprocity law itself proved by Michael Harris and Richard Taylor in was reformulated within the context of class field 1998. A second proof was given by Guy Henniart theory. This theory was initiated by Kronecker in shortly thereafter. The Local Langlands Corre- the case of quadratic imaginary fields. It was de- spondence is a nonabelian generalization of the rec- veloped into a general framework by Weber and iprocity law of local class field theory. It gives a re- Hilbert in the 1890s and was proven by Furtwan- markable relationship between nonabelian Galois gler, Takagi, and Artin in the first quarter of the groups and infinite-dimensional representation twentieth century. theory. Posed as an open problem thirty years ago In simple terms, class field theory seeks to de- in [L1], its proof in full generality represents a scribe all of the finite abelian extensions of a num- milestone in algebraic number theory. The goal of ber field F, that is, the finite Galois extensions this note, aimed at the nonspecialist, is to state the K/F such that Gal(K/F) abelian.1 The theory ac- main result with some motivations and necessary complishes its goal by establishing a deep relation definitions. between generalized ideal class groups attached to F and decomposition laws for prime ideals in Reciprocity Laws abelian extensions K/F. This theory is elementary For any positive integer d, the law of quadratic rec- in the case F = Q. According to the Kronecker- iprocity describes the primes p for which the con- Weber theorem, every abelian extension of Q is con- gruence x2 d mod p has a solution. It implies, tained in a cyclotomic extension Q(e2i/N) for quite counterintuitively, that the existence of a so- some integer N. On the other hand, the general- lution to this congruence modulo p depends only ized ideal class groups of Q are just the multi- on the residue class of p modulo 4d. Despite the plicative groups (Z/NZ) and their quotients. Class large number of different proofs, it remains one field theory yields the isomorphism between of the deepest and most mysterious results of el- (Z/NZ) and the Galois group of K = Q(e2i/N) in ementary number theory. which a residue class m ∈ (Z/NZ) maps to the Ga- 2i/N 2im/N The search for generalizations of quadratic rec- lois automorphism sending e to e . Im- iprocity, beginning with work of Gauss and Eisen- plicit in this isomorphism is the decomposition law: stein on cubic and higher reciprocity laws, moti- if p is a prime not dividing N and f is the order of vated a great deal of number-theoretic research in p in (Z/NZ) , then (p) factors in K as a product of the nineteenth century. However, it was not pos- 1Class field theory treats all global fields, i.e., finite ex- Jonathan Rogawski is professor of mathematics at the Uni- tensions of Q or of the field Fq(X) of rational functions versity of California at Los Angeles. His e-mail address is over a field of q elements. In this exposition, we will dis- [email protected]. cuss only the number field case.

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ϕ(N)/f distinct prime ideals in the ring of integers extensions of a local field F, i.e., a locally compact in K. Concretely, this means that xN 1 factors nondiscrete field. In characteristic zero, the only modulo p as a product of ϕ(N)/f distinct irre- such fields are R, C, and finite extensions of the ducible polynomials and, in particular, the num- field Qp. The latter examples are called p-adic ber of factors in the factorization modulo p de- fields. For a local field F, the reciprocity law takes pends only on p modulo N. The law of quadratic a similar form, with the multiplicative group F tak- reciprocity follows from this fact in a natural way. ing the place of the idele group. More precisely, the For an arbitrary number field F, the corre- local reciprocity law is a continuous homomor- sponding assertion is the Artin reciprocity law. It phism is most cleanly stated using “ideles” in place of ab ideals and a maximal abelian extension Fab of F r : F → Gal(F /F). in place of finite abelian extensions. Here we work As in the global number-field case, the map r pro- inside a fixed algebraic closure F¯ of F and take Fab vides tangible nontrivial information about the to be the union of all finite abelian extensions K arithmetic object Gal(Fab/F) in terms of the ele- of F in F¯. The “idele group” of F is defined in mentary object F. If F is p-adic, r is injective and terms of the inequivalent completions of F, each its image is dense, but it is no longer surjective. In of which is defined by a “place”. Consider first the particular, r extends by continuity to an identifi- case F = Q. The places v are the primes cation of Gal(Fab/F) with the profinite comple- p =2, 3, 5,... and one infinite place v = ∞. The tion of F, and thus we obtain a simple descrip- completions relative to these places are the fields ab tion of Gal(F /F). For example, if F = Qp, where Q of p-adic numbers for v finite and Q∞ = R for p p is an odd prime, then Q is isomorphic to v = ∞. For p prime let Z be the ring of p-adic in- p p ab Z (Z/p) Zp and Gal(Q /Qp) is isomorphic tegers within Qp. The adele ring AQ is the subring p Zb Z Z Z Zb of tuplesQ (xv )=(x∞,x2,x3,x5,...) in the direct to ( /p ) p, where is the profinite com- Q Z product v v (product over all v) such that xv is pletion of . Of course, local class field theory in Zv for all but at most finitely many finite places consists of much more than an abstract group iso- v. The idele group is the group AQ of invertible el- morphism. See [S] for a list of the properties of r ab ements in AQ. Locally compact topologies [CF] may relating the structure of F to the arithmetic of F . be defined for AQ and AQ so that AQ is a topolog- Nonabelian Reciprocity ical ring and AQ is a topological group. The defi- nitions are such that the diagonal embedding In the second of his Two Lectures on Number The- x 7→ (x, x, x, x, ...) identifies Q and Q with discrete ory, Past and Present ([W], vol. III, p. 301), André subgroups of AQ and AQ , respectively. Weil summed up as follows the state of affairs in For an arbitrary number field, there are finitely the 1920s and 1930s after the main results of many infinite places v, for which Fv is R or C, and class field theory had been established: infinitely many finite places, for which Fv is a fi- Artin’s reciprocity law, which in a sense nite extension of Qp for some prime p. The adele contains all previously known laws of ring AF and idele group AF are defined in terms reciprocity as special cases, deals with of the localizations Fv as in the case F = Q. Again, a strictly commutative problem. It es- the respective diagonal embeddings of F and F tablishes a relation between the most have discrete images. general extension of a number-field The Artin reciprocity law is a certain continu- with a commutative Galois group on ous, surjective homomorphism the one hand, and on the other hand the multiplicative group over that field. r : A /F → Gal(Fab/F) F Where do we go from there? Well—of defined explicitly in terms of “Frobenius auto- course we take up the noncommuta- morphisms”. It is from this explicit definition of tive case. r that it is possible (with some work) to deduce fun- The group Gal(Fab/F) is equal to Gal(F/F¯ )ab, i.e., damental number-theoretic information about F. the quotient of Gal(F/F¯ ) by the closure of its com- In theQ case F = Q, AQ /Q can be identified with mutator subgroup. A noncommutative general- R Z + Q p p , and r amounts to the identification ization of Artin’s law would, at the least, provide Z Qab Q of p p with Gal( / ) coming from the con- a description of the full Galois group Gal(F/F¯ ) in struction of Qab by adjoining all roots of unity. This some way that expresses the decomposition law formulation puts together all of the class field the- for primes in finite extensions of F. It was a major ory isomorphisms for the extensions Q(e2i/N) in obstacle, however, to find the right language for one package. fomulating such a law. One gets a sense of the dif- The theory just described is called global class ficulty of the problem from a remark of Weil2 to field theory because it deals with a number field. Local class field theory is concerned with abelian 2Note [1971a] in [W], vol. III, p. 457.

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the effect that E. Noether, E. Artin, and H. Hasse The kernel of the map → m is a closed sub- had hoped in vain that their theory of simple al- group of Gal(F/F¯ ) called the inertia subgroup IF of gebras would lead to a nonabelian theory but that F. An element mapping to m =1is called a Frobe- by 1947 Artin confided that he was no longer sure nius automorphism. In any case, we have a (non- 3 that such a theory existed. The breakthrough canonical) isomorphism W (F/F¯ ) ' Z IF , and we came out of Langlands’s discovery in the 1960s of use this to make W (F/F¯ ) into a topological group, the conjectural “Principle of Functoriality”, which taking the product of the discrete topology on Z included a formulation of a nonabelian reciproc- and relative topology from Gal(F/F¯ ) on IF. With this ity law (both global and local) as a special case. definition, the restriction r 0 of the reciprocity map It turns out that in the nonabelian case, a key induces an isomorphism of topological groups shift of viewpoint is necessary: one describes the 0 representations of Gal(F/F¯ ) rather than Gal(F/F¯ ) (1) r : F → W (F/F¯ )ab. itself. Thus, the global reciprocity law is formulated as a general conjectural correspondence between The Dual Perspective Galois representations and automorphic forms. In fact, in the 1960s Serre and Shimura derived non- As mentioned above, the key to the nonabelian rec- abelian reciprocity laws of this type using the co- iprocity law is to dualize the reciprocity isomor- homology of modular curves. However, the Lang- phism (1). Thus, for F a p-adic field, we consider 6 lands reciprocity law, like Artin’s abelian law, posits the isomorphism of character groups a correspondence independent of any particular (2) 1 : Hom(F , C ) → Hom(W (F/F¯ ), C ) geometric construction. 0 The Principle of Functoriality has led more gen- induced by r . erally to a web of interrelated results and conjec- Since the nonabelian generalization of a char- tures which together make up the Langlands Pro- acter is an irreducible representation, it is rea- gram. This program has both a global and local sonable to consider the set Gn of equivalence aspect. See [A], [G], [L2], [K], [R] and the references classes of irreducible n-dimensional representa- they contain for a general overview of this program tions of W (F/F¯ ) for all n 1. Then and its consequences.4 The article [D] mentions the G ¯ C relation between the Langlands Program and the 1 = Hom(W (F/F), ), modularity of elliptic curves. and so from this point of view, the generalization of (2) would be some map of the form The Weil Group To describe the local nonabelian reciprocity law, n : ??? →Gn we introduce the Weil group W (F/F¯ ).5 Assume that F is a p-adic local field, i.e., a finite extension of with target Gn. What is much less obvious is the Qp for some prime p. Let the ring of integers be correct source for n . OF. This ring has a unique maximal ideal, neces- It turns out that the source of n is the set of sarily principal, and we let $ be any generator, a equivalence classes of supercuspidal representa- so-called prime element. Denote by q the cardinality tions of GLn(F), which are a type of infinite-di- of the residue field k = O /($). One knows that F F mensional representation that we define below. The a Galois automorphism ∈ Gal(F/F¯ ) induces an global version relates irreducible n-dimensional automorphism ¯ of the residue field k of F¯, and F representations of the Galois group and cuspidal the map → from Gal(F/F¯ ) to Gal (kF /kF ) is sur- representations of the group GLn(AF ). It is more jective. The Galois group Gal(k /k ) is isomorphic F F difficult to state precisely [CL], [L1], [R]. to Zb and the Weil group W (F/F¯ ) is defined as the In hindsight it is not surprising that the non- dense subgroup of Galois automorphisms such m abelian theory was not developed until the 1960s. that ¯ is of the form x → xq for some integer m. The theory of infinite-dimensional representations of reductive groups was first developed actively in 3Coincidentally, 1947 was the year that Harish-Chandra received his Ph.D. in physics under P. Dirac from Cam- the 1950s, and the p-adic case was not studied in- bridge University. See [A] elsewhere in this issue of the No- tensively until the 1960s. The existence of super- tices for a discussion of Harish-Chandra’s work and its cuspidal representations for GL2 over a p-adic relation with the Langlands Program. field was apparently first observed by F. Maut- 4For historical documents and commentary on functori- ner.7 They were first studied in greater generality ality as well as Langlands’s papers in downloadable in the work of Jacquet and Harish-Chandra. form, visit the Web site http://sunsite.ubc.ca/ DigitalMathArchive/Langlands/. 6Here and below we consider only continuous charac- 5See Tate’s article in [CO], Part II, for an excellent discussion ters, but omit the continuity in the notation. of Weil groups in the local and global cases. 7According to an oral communication from J. Shalika.

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Admissible Representations if we begin with the global case. The global factors arose first historically, and it was an important the- Let G = GLn(F), and let (,V) be a representation of G on a complex vector space V. Since F is a p- oretical step to define their local counterparts. For adic field, G has a family of open compact sub- the moment then F will denote a number field. groups {Km}m1 , where Km consists of the inte- First of all, we can follow Hecke and attach an gral matrices g ∈ G such that g 1 mod ($m). L function L(s,) to each continuous homomor- A → C We say that (,V) is admissible if phism : F /F . Such a homomorphism is • every vector v ∈ V is fixed by Km for some m, called a Hecke character. A Hecke character gives and rise, by restriction, to a homomorphism → C • the subspace of vectors fixed by each Km is v : Fv for all places v of F, and the Hecke finite-dimensional. L function of is defined as an infinite product From now on (,V) will denote an irreducible Y admissible representation. We note that to each ir- L(s,)= L(s,v ) reducible unitary representation ( 0,V0) there is v attached an admissible representation (,V) in a of local factors L(s,v ). 0 natural way: V is the subspace of vectors v ∈ V If v is finite, we say that v is unramified if the O O fixed by Km for some m, and is the restriction restriction of v to v is trivial, where v is the 0 of to V. Furthermore, the space V of an irre- group of units in the ring Ov of integers in Fv. In s 1 ducible admissible representation is either one-di- this case, we set L(s,v )=(1 ($v )qv ) , where mensional or infinite-dimensional. Indeed, if $v is any prime element in Ov and qv = |Ov /($v )|. ∞ dim V< , then Km is contained in ker for some If v is finite but v is ramified, then L(s,v )=1. m, and since ker is normal, it must contain If v is infinite, L(s,v ) is defined in terms of a cer- SLn(F). The contragredient of (,V) is the repre- tain finite product of gamma functions. The con- sentation of G on the admissible dual space tinuity of implies that v is unramified for all V , that is, the space of linear functionals on V that but finitely many v, and thus L(s,) is an Euler are fixed by Km for some m. A matrix coefficient product. The product converges absolutely and of (,V) is a function on G of the form uniformly for Re(s) sufficiently large, and it thus f (g)=h(g)v,wi with v ∈ V and w ∈ V ; here defines an analytic function in a suitable right half hv,wi is the bilinear pairing on V V . The rep- plane. The Riemann zeta function is the Hecke L resentation is called supercuspidal if the support function corresponding to the trivial Hecke char- of every matrix coefficient is compact modulo the acter of AQ /Q .8 The Dirichlet L functions that center Z of G. arise in the proof that there are infinitely many The Local Langlands Conjecture asserts that for primes in arithmetic progressions are also Hecke all n, there exists a natural bijection L functions, apart from a contribution from a fi-

n : Cn →Gn , nite product arising from the infinite places. Hecke proved that his L functions L(s,) can be where Cn is the set of equivalence classes of su- meromorphically continued to the entire complex percuspidal representations of G = GLn(F). In the plane and that they satisfy a global functional special case n =1, we have C1 = Hom(F , C ). Of equation of the form course, to give this conjecture some content, we 1 must specify what is meant by “natural”. To do so, L(s,)=(s,)L(1 s, ), we exploit the fact it is possible to attach so-called where (s,) is a factor of the form A Bs for some L and factors to objects in either C or G , al- n n constants A and B. This generalizes the functional though the procedure is quite different in the two equation of the Riemann zeta function. An im- cases. portant method for proving Hecke’s results using L and Factors Fourier analysis on local fields and adele rings was developed in 1950 by J. Tate in his well-known L factors generalize the individual factors of the Ph.D. thesis and also by K. Iwasawa. The great Euler product representation of the Riemann zeta virtue of this method is that it provides a local in- function (s), which is represented for Re(s) > 1 terpretation of the local factors and, equally im- by the Euler product Y portant, a factorization of (s,) into local fac- (s)= (1 p s ) 1. tors as described below. p What emerges clearly in the work of Langlands and Jacquet-Langlands is that the nonabelian There are two fundamentally different ways to make the generalization, both of which play a key 8More precisely, it is the product of the Riemann zeta func- role in our story. Our goal is to define more gen- tion and a factor coming from v = ∞. This is the gamma eral L factors and the closely related factors in factor that enters the usual functional equation for the the local case, but our description will be clearer zeta function.

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generalization of a Hecke character is a cuspidal of a number field F . Every such factors representation [BK], [CO], [G], [L1] of the adelic through Gal(K/F) for some finite Galois extension group GLn(AF ) or, more generally, of the adelic K/F. For each finite place v, the choice of an em- points of any reductive group over F. Just as a bedding F¯ → F¯v induces an embedding Hecke character has local components v , a cus- Gal(F¯v /Fv ) → Gal(F/F¯ ), uniquely defined up to pidal representation has local components v conjugation. The pullback v of to Gal(F¯v /Fv ) is that are irreducible admissible representations of thus well defined up to equivalence, and we may GLn(Fv ). In this light, Tate’s thesis concerns the define a local factor case n =1, and one can then ask if there exists a s | Iv 1 generalization of it to n>1. In the late 1950s and L(s,v ) = det(I qv v (v ) V ) . early 1960s the analogous problem for the multi- Here V Iv denotes the subspace of invariants under plicative group of a division algebra in place of GL n the inertia subgroup I of F , and is a Frobe- was considered by Godement and Tamagawa. v v v nius element. For infinite v the local factor is a fi- Shimura also considered the quaternion algebra nite product of gamma functions, as in the case of case in connection with the analytic continuation Hecke L functions. The Artin L function is defined of the zeta-functions attached to arithmetic quo- Q as an infinite product L(s,)= L(s, ). tients of the upper half-plane (Shimura curves). A v v The infinite product defines L(s,) as a mero- full representation-theoretic treatment of the prob- morphic function for Re(s) > 1. R. Brauer proved lem for GL containing, in particular, a detailed n that L(s,) extends meromorphically to the com- local theory was given by Godement and Jacquet. plex plane and satisfies a functional equation of We shall focus only on the local part of the the- the form L(s,)=(s,)L(1 s,ˆ), where ˆ is the ory that is relevant to the statement of the Local contragredient of . The existence of a factoriza- Langlands Correspondence. Let F be a p-adic field. tion (s,) as a product of local factors was The Godement-Jacquet theory attaches local L and factors to any irreducible admissible represen- established by Langlands using local methods (unpublished, but now available on the Web site tation (,V) of GLn(F) in the following way. One introduces the so-called zeta integrals: in footnote 4). Deligne published a shorter Z proof using global methods. The factorization, Z(s,f,)= f (g)(g)|det(g)|s dg which depends on the choice of a nontrivial addi- A → C GLn(F) tive characterQ : F /F , has the form (s,)= v (s,v ,v ) in analogy with the fac- where dg is a Haar measure on GLn(F), f is a ma- torization provided by the theory of Tate and trix coefficient of , and is a locally constant Godement-Jacquet. The local factor (s,v ,v ) de- function of compact support on the vector space pends only on the restriction v of to Fv and Mn(F) of n-by-n matrices over F. Although the in- is characterized by suitable compatibility re- tegral converges for Re(s) sufficiently large, it can quirements. be shown that Z(s,f,) extends to a rational func- tion in q s. Furthermore, there is a unique zeta in- The Local Langlands Correspondence s 1 tegral of the form L(s,)=P(q ) , where P is a Once the local factors are known to exist, it is polynomial of degree at most n with P(0) = 1, such tempting to define the map n by requiring that that Z(s,f,)/L(s,) is entire for all and f. Fi- L(s,)=L(s,n()) and (s,)=(s,n()) for nally, the choice of a nontrivial additive character all supercuspidal . Unfortunately, this does not : F → C plays a role: one proves that for each determine n uniquely. Henniart proved, however, choice of there is a unique factor (s,,) such that n can be characterized by matching a more that the following local functional equation holds general type of local factors. Thus we introduce the for all f and : final ingredient needed in order to state the cor- respondence precisely: the local factors attached Z(s,f,) Z(1 s,f , b) = (s,,) . to pairs. Again let F be a p-adic field. L(s,) L(1 s, ) 0 If and are representations of Gal(F/F¯ ) of 0 Here f (g)=f (g1), b is the Fourier transform of dimensions n and n respectively, the results relative to the character of F, and is the quoted in the previous paragraph allow us to at- tach local factors L(s, 0) and (s,, 0) to contragredient representation of . Furthermore, 0 s the tensor product , which is a representa- (s,,) is of the form q , where and are 0 constants. tion of dimension nn . The analogue of this con- The second generalization of the Riemann zeta struction for admissible representations was car- function is due to E. Artin, who defined an L func- ried out by Jacquet, Piatetski-Shapiro, and Shalika. They defined local factors L(s, 0) and tion L(s,) for any (continuous) finite-dimensional 0 0 representation of the Galois group (s,, ) for any pair and of irreducible admissible representations of GLn(F) and GLn0 (F) → C : Gal(F/F¯ ) GLn( ) respectively. These reduce to the Godement-Jacquet

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factors when 0 is the trivial representation of tions of the appropriate type, the modular curves GL1(F). provide natural candidates for 2(). The difficulty The center Z of GLn(F) consists of scalar ma- lies in showing that all of these candidates coin- trices and thus is isomorphic to F. If is an ir- cide and satisfy the conditions of the Main Theo- reducible representation of GLn(F), then there is rem. a character ω of Z, called the central character One would like to employ the same strategy in of , such that (z)=ω (z)I for all z ∈ Z. We re- the case n>2. However, there are no analogues gard ω as a character of F . of the modular curves for GLn in this case. The way around this, according to a suggestion first made Main Theorem. There exists a unique family of bi- by H. Carayol, is to exploit the fact that the group jections GLn over a p-adic field can be obtained as the lo- C →G n : n n calization at a p-adic place of certain “twisted” unitary groups over a suitable totally real number for n 1 satisfying the following conditions: 0 field F . The unitary groups do give rise to a fam- (a) is the correspondence (2) of abelian local 1 ily of “Shimura varieties” that are analogous to the class field theory. 0 family of modular curves, and one obtains candi- (b) For all pairs ∈Cn and ∈Cn0 , we have dates for the representations n() as in the case 0 0 L(s, )=L(s,n() n0 ( )) n =2. Carrying out this strategy is a difficult task. We and 0 0 refer to [C] for an excellent technical overview of 0 (s, )=(s,n() n ( )). the strategy of the proofs. The articles [C] and (c) For all , 1(ω ) = det(n()). [Ku] describe also how the Main Theorem can be extended to a correspondence between the set of As mentioned above, the uniqueness assertion equivalence classes of all irreducible admissible had been established earlier by Henniart (1993). The representations and the set of finite-dimensional articles [HT] and [H] establish existence in full gen- representations of the so-called Weil-Deligne group. erality, relying heavily on global results and meth- Many cases of the Local Langlands Correspon- ods coming from the theory of Shimura varieties dence had been established previously. We men- and automorphic forms. Prior to that, Harris had tion only the case n =2which, of course, was han- proved the local correspondence for n

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breakthrough, Laurent Lafforgue succeeded in gen- [R] J. ROGAWSKI, Functoriality and the Artin Conjecture, eralizing Drinfeld’s approach and established the in [BK], pp. 331–353. “global Langlands correspondence” in full gener- [S] J.-P. SERRE, Local Fields, Graduate Texts in Math., vol. 67, Springer-Verlag, New York and Berlin, 1979. ality for GLn over a global field of positive char- [W] A. WEIL, Oeuvres Scientifiques, Springer-Verlag, New acteristic. York-Heidelberg-Berlin, 1979. References [A] J. ARTHUR, Harmonic analysis and group represen- tations, Notices Amer. Math. Soc. 47 (January 2000). [BK] T. N. BAILEY AND A. W. KNAPP, eds., Representation The- ory and Automorphic Forms, Instructional Confer- ence, International Centre for Mathematical Science, March 1996, Edinburgh, Scotland, Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1996. [C] H. CARAYOL, Preuve de la conjecture de Langlands lo- cale pour GLn; travaux de Harris-Taylor et Hen- niart, Séminaire Bourbaki, 51ème année, 1998-99, March 1999. [CF] J. W. S. CASSELS AND A. FRÖHLICH, eds., Algebraic Num- ber Theory, Proceedings of the instructional con- ference held at the University of Sussex, Brighton, September 1–17, 1965, Academic Press, London and New York, 1986. [CL] L. CLOZEL, Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic Forms, Shimura Varieties, and L-Functions, Vol. I, Perspec- tives in Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159. [CO] A. BOREL AND W. CASSELMAN, eds., Automorphic Forms, Representations and L-Functions, Proceedings of the Twenty-fifth Summer Research Institute, Confer- ence at Oregon State University, Corvallis, OR, July/August 1977, Proc. Sympos. Pure Math., vol. 33, parts 1 and 2, Amer. Math. Soc., Providence, RI, 1979. [D] H. DARMON, A proof of the full Shimura-Taniyama- Weil conjecture is announced, Notices Amer. Math. Soc. 46 (December 1999), 1397–1401. [G] S. GELBART, An elementary introduction to the Lang- lands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), 177–219. [HT] M. HARRIS and R. TAYLOR, On the geometry and co- homology of some simple Shimura varieties, preprint (1998). [H] G. HENNIART, Une preuve simple des conjectures de Langlands pour GLn sur un corps p-adique, preprint (January 1999). [J] H. JACQUET, Principal L-functions for GL(n), in [BK], pp. 321–329. [K] A. KNAPP, Introduction to the Langlands program, in [BK], pp. 245–302. [Ku] STEPHEN S.K. UDLA, The local Langlands correspon- dence: The non-Archimedean case, Motives, Proc. Sympos. Pure Math., vol. 55, part 2, Amer. Math. Soc., Providence, RI, 1994, pp. 365–391. [L1] R. LANGLANDS, Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applica- tions, III, Lecture Notes in Math., vol. 170, Springer, Berlin, 1970, pp. 18–61. [L2] ———, L-functions and automorphic representations, Proceedings of the International Congress of Math- ematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp.165–175.

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