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Langlands program

Contents

Idea

What is called the Langlands correspondence in number theory (Langlands 67) is first of all a conjectural correspondence (a bijection subject to various conditions) between

  1. nn-dimensional complex linear representations of the Galois group Gal(F¯/F)Gal(\bar F/F) of a given number field FF, and

  2. certain representations – called automorphic representations – of the nn-dimensional general linear group GL n(𝔸 F)GL_n(\mathbb{A}_F) with coefficients in the ring of adeles of FF, arising within the representations given by functions on the double coset space GL n(F)\GL n(𝔸 F)/GL n(𝒪)GL_n(F) \backslash GL_n(\mathbb{A}_F)/GL_n(\mathcal{O}) (where 𝒪= v𝒪 p\mathcal{O} = \prod_v \mathcal{O}_p is the ring of integers of all formal completions of FF).

This is motivated from the abelian case (n=1n=1), which is fully understood: For n=1n = 1 then an nn-dimensional representation of the Galois group factors through GL 1GL_1 and hence through an abelian group. Therefore, by adjunction, it is equivalently a representation of the abelianization of the Galois group. The Kronecker-Weber theorem says that for F=F = \mathbb{Q}, then the abelianized Galois group is the idele class group GL 1()\GL 1(𝔸)GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A}), and hence 1-dimensional representations of the Galois group are equivalently representations of this. Moreover, one finds that for any prime number pp, then those representations which are “unramified at pp” are invariant under the subring of p-adic integers, hence are representations of the double quotient group GL 1()\GL 1(𝔸)/GL 1( p)GL_1(\mathbb{Q}) \backslash GL_1(\mathbb{A})/GL_1(\mathbb{Z}_p). More generally, the Artin reciprocity law says that for number fields there is an isomorphism between GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪 K)GL_1(K) \backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O}_K) and the abelianized Galois group.

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Moreover:

Conjecture 1 To each such automorphic representation π\pi is associated an L-function – the automorphic L-function L πL_\pi – and in generalization of Artin reciprocity the conjecture of Langlands is that the Artin L-function L σL_\sigma associated with the given Galois representation σ\sigma is equal to this: L σ=L πL_\sigma = L_\pi (Gelbhart 84, conjecture 1 (page 27 (204))).

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More generally, analogous statements are supposed to hold for general reductive algebraic groups GG other than GL nGL_n. Here now an L-function is assigned to data which in addtion to the Galois representation consists of a linear representation LGGL n{}^L G \longrightarrow GL_n of the Langlands dual group of GG.

First of all:

Conjecture 2 This more general L-function is conjectured to indeed behave like a decent L-function in that it has meromorphic analytic continuation to the complex plane and satisfies the “functional equation”-invariance under sending its parameter ss to 1s1-s (Gelbhart 84, conjecture 2’ (page 29 (205))).

Second:

Conjecture 3 This construction is supposed to behave well with respect to an analytic homomorphism LG LG {}^L G \to {}^L G^{'} in that when changing the representation of LG {}^L G^{'} by precomposition with this homomorphism one may find an accompanying change of Galois representation/automorphic representation from GG to G G^{'} such that the associated L-function remains invariant under these joint changes. This statement is what Robert Langlands calls functoriality (Gelbhart 84, conjecture 3 (page 31 (207)))

In fact this last conjecture implies the previous two (Gelbhart 84, (page 32 (208))).

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Various versions and refinements of this conjecture have since been considered, for some perspective see (Taylor 02, Langlands 14, Harris 14). On the one hand the “localization” of the program to local fields leads to the conjecture of local Langlands correspondences (Gelbhart 84, (page 34 (210))). On the other hand, the interpretation of the above story dually in arithmetic geometry in view of the function field analogy motivates the conjectural geometric Langlands correspondence, based on the following analogy:

analogies in the Langlands program:

arithmetic Langlands correspondencegeometric Langlands correspondence
ring of integers of global fieldstructure sheaf on complex curve Σ\Sigma
Galois groupfundamental group of Σ\Sigma
Galois representationflat connection/local system on Σ\Sigma
idele class group mod integral adelesmoduli stack of line bundles on Σ\Sigma
nonabelian \;moduli stack of vector bundles on Σ\Sigma
automorphic representationHitchin connection D-module on bundle of conformal blocks over the moduli stack

From this arithmetic geometry point of view the Langlands conjecture seems to speak of a correspondence that sends Dirac distributions on the moduli space of flat connections over an algebraic curve to certain “automorphic” functions on the moduli stack of bundles on the same curve. This suggests that the Langlands correspondence should be understood as a nonabelian version of a Fourier-Mukai-type integral transform. This version of the conjecture is known as the geometric Langlands correspondence. See there for more details.

References

The original conjecture is due to

Surveys of the state of the program include

Introductions and expository surveys include

  • Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)

  • Edward Frenkel, Commentary on “An elementary introduction to the Langlands Program” by Steven Gelbart, Bull. Amer. Math. Soc. 48 (2011), 513-515, (pdf)

  • Mark Goresky, Langlands’ conjectures for physicists (pdf)

  • Minhyong Kim, A superficial introduction to Langlands functoriality, 2014 (pdf)

Discussion with an eye towards geometric class field theory and geometric Langlands duality and quantum field theory is in

  • Mikhail Kapranov, Analogies between the Langlands Correspondence and Topological Quantum Field Theory, in Functional Analysis on the Eve of the 21st Century Progress in Mathematics Volume 131/132, 1995, pp 119-151

  • Peter Toth, Geometric abelian class field theory, 2011 (web)

  • Edward Frenkel, Gauge Theory and Langlands Duality (arXiv:0906.2747)

More resources are at

Revised on December 13, 2014 21:25:50 by Urs Schreiber (127.0.0.1)