Langlands program

The Langlands Program of Mathematics is a series of far-reaching conjecture that the number theory and representation theory of groups linked together. They have been set up by Robert Langlands since 1967 .

Connection to number theory

Artin's law of reciprocity , which generalizes the quadratic reciprocity law , can be seen as the starting point of the program . Artins reciprocity law associates a algebraic number field whose Galois over commutative is (abelian), an L-function of the one-dimensional representations of the Galois to and states that these L-function with a certain Dirichlet L-row matches. ${\ displaystyle \ mathbb {Q}}$

For non-Abelian Galois groups and higher-dimensional representations one can also define L-functions in a natural way.

Automorphic representations

Langlands' idea was to find a suitable generalization of Dirichlet's L-functions that would allow Artin's statement to be formulated in this more general framework.

Hecke had already associated Dirichlet's L-functions with automorphic forms , i.e. with holomorphic functions of the upper half-plane of complex numbers that satisfy certain functional equations (see Hecke operator ). Langlands generalized this to automorphic cuspid representations . These are infinite-dimensional irreducible representations of the general linear group above the ring of Adele of , whereby this ring takes into account all completions of , see p-adic numbers . ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle \ mathrm {GL} _ {n}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {Q}}$

Langlands assigned certain L-functions to these automorphic representations and assumed that every L-function of a finite-dimensional representation of the Galois group corresponds to the L-function of an automorphic cuspidual representation. This is the so-called “reciprocity presumption”.

A general principle of functoriality

Langlands generalized this even further: Instead of the general linear group , one can consider other reductive groups . Langlands constructed a complex Lie group for such a group , and for every automorphic cuspid representation of and every finite dimensional representation of he defined an L-function. One of his conjectures then says that these L-functions satisfy certain functional equations that generalize those of known L-functions. ${\ displaystyle \ mathrm {GL} _ {n}}$ ${\ displaystyle G}$ ${\ displaystyle {} ^ {\ mathrm {L}} G}$ ${\ displaystyle G}$ ${\ displaystyle {} ^ {\ mathrm {L}} G}$

In this context, Langlands formulated a general “functoriality principle”: If two reductive groups and a morphism between their L-groups are given, according to this assumed principle, their automorphic representations are connected to one another in a way that is compatible with their L-functions. This functoriality implies all other conjectures. It is of the type of the construction of an induced representation , what was called a "lifting" in the traditional theory of automorphic forms. Attempts to specify such a construction directly have only led to limited results.

All of these assumptions can also be made for other bodies . Instead of you can use algebraic number fields , local fields and function fields , i. H. consider finite field extensions of , where a prime number and denotes the field of the rational functions over the finite field with elements. ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {F} _ {p} (t)}$ ${\ displaystyle p}$ ${\ displaystyle \ mathbb {F} _ {p} (t)}$ ${\ displaystyle p}$

Ideas that led to the Langlands program

In the program the following ideas were received: the philosophy of tip shapes, which a few years earlier by Israel Gelfand had been formulated, access by Harish-Chandra to semisimple Lie groups and in the technical sense, the trace formula of Selberg and others. The novelty in Langland's work, besides the technical depth, was the presumed direct connection to number theory and the functional structure of the whole.

In the work of Harish-Chandra, for example, one finds the principle that what can be done with a semi-simple (or reductive) Lie group should be done for everyone. Thus, once the role of low-dimensional Lie groups such as that in the theory of modular forms had been recognized, the way was open to speculation about anything . ${\ displaystyle \ mathrm {GL} (2)}$ ${\ displaystyle \ mathrm {GL} (n)}$ ${\ displaystyle n> 2}$

The idea of the tip shape came from the tips in module curves , but it was also visible in spectral theory as a discrete spectrum , in contrast to the continuous spectrum of Eisenstein series . This relationship becomes technically far more complicated for larger Lie groups, since the parabolic subgroups are more numerous.

Results and prices

Parts of the local body program ended in 1998 and the functional body program ended in 1999. Laurent Lafforgue received the Fields Medal in 2002 for his work on functional bodies . This continued earlier research by Vladimir Drinfeld , who was also awarded the Fields Medal in 1990. The program has only been proven for number fields in a few special cases, in part by Langlands itself. For local function fields, the Langlands conjecture was proven by Gérard Laumon , Michael Rapoport , Ulrich Stuhler . The local long-country conjecture (for local p -adic bodies) was proven in 1998 by Michael Harris and Richard Taylor and independently by Guy Henniart .

Langlands received the Wolf Prize for Mathematics in 1996 for his work on these conjectures and the Abel Prize in 2018 . For proof of the fundamental lemma , Ngô Bảo Châu received the Fields Medal in 2010.

Geometric Langlands program

Because of the great difficulties in realizing the Langlands program in number theory, some mathematicians ( Alexander Beilinson , Vladimir Drinfeld , Gérard Laumon from the 1980s, Edward Frenkel , Dennis Gaitsgory , Kari Vilonen ) have switched to function fields instead of number fields in Langlands correspondence (Curves over complex numbers or finite fields). This follows an old tradition, instead of studying the more difficult case of number fields first of all the simpler case of function fields. The area has links to string theory and conformal quantum field theories since the work of Anton Kapustin and Edward Witten , who linked S-duality with the geometric Langlands correspondence. There are also connections to topological quantum field theory .

literature

Stephen Gelbart : An Elementary Introduction to the Langlands Program. In: Bulletin of the AMS. Volume 10, 1984, pp. 177-219, ams.org
Anthony W. Knapp : Introduction to the Langlands program. In: TN Bailey, AW Knapp (Ed.): Representation theory and automorphic forms. In: Amer. Math. Soc. 1997, pp. 245-302.
AW Knapp: Group Representations and Harmonic Analysis from Euler to Langlands. Part 1 (PDF; 183 kB), Part 2 (PDF; 177 kB). In: Notices AMS. 1996.
Solomon Friedberg : What is the Langlands Program? In: Notices AMS , June / July 2018, ams.org

Geometric Langlands program:

Edward Frenkel : Lectures on the Langlands Program and Conformal Field Theory. arxiv : hep-th / 0512172 .
E. Frenkel: Langlands program, trace formulas, and their geometrization. In: Bull. Amer. Math. Soc. Volume 50, 2013, pp. 1-55, ams.org
E. Frenkel: Gauge theory and the Langlands duality. Bourbaki Seminar 2009, arxiv : 0906.2747 .

Web links

Individual evidence

^ G. Laumon, M. Rapoport, U. Stuhler: -elliptic sheaves and the Langlands correspondence. In: Invent. Math. 113: 217-338 (1993). ${\ displaystyle {\ mathcal {D}}}$

^ A. Kapustin, E. Witten: Electric-Magnetic Duality And The Geometric Langlands Program. In: Communications in Number Theory and Physics. Volume 1, 2007, pp. 1-236, arxiv : hep-th / 0604151 .

[1] G. Laumon, M. Rapoport, U. Stuhler: -elliptic sheaves and the Langlands correspondence. In: Invent. Math. 113: 217-338 (1993). ${\ displaystyle {\ mathcal {D}}}$

[2] A. Kapustin, E. Witten: Electric-Magnetic Duality And The Geometric Langlands Program. In: Communications in Number Theory and Physics. Volume 1, 2007, pp. 1-236, arxiv : hep-th / 0604151 .