comm-rogawski.qxp 11/12/99 1:15 PM Page 35 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. JANUARY 2000 NOTICES OF THE AMS 35 comm-rogawski.qxp 11/12/99 1:15 PM Page 36 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.