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Langlands program

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Idea

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

  1. nn-dimensional 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 conjecture is motivated from the following special case, 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.

Various versions and refinements of this conjecture have since been considered, for some perspective see (Taylor 02, Langlands 14, Harris 14).

In particular, interpretation of the above story dually in arithmetic geometry has led to some developments. Namely under the function field analogy we have that

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

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

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

More resources are at

Revised on August 18, 2014 21:23:51 by David Corfield (46.208.114.209)