Langlands Program

In mathematics, the Langlands program is a network of far-reaching and influential hypotheses about the connections between number theory and geometry . It was proposed by Robert Langlands in 1967 and 1970. It seeks to connect Galois groups in algebraic number theory with automorphic forms and the theory of representations of algebraic groups over local fields and adeles . The Langlands program, widely known as the largest project in modern mathematical research, was described by Edward Frenkel as “the theory of the great unification of mathematics” ^[1] .

Langlands received the Abel Prize for the Langlands program in 2018.

Content

Context

The Langlands program is based on previously developed ideas: the philosophy of parabolic forms , formulated several years earlier by Harish-Chandra and Israel Gelfand in 1963, the work of Harish-Chandra on semisimple Lie groups , and in technical terms - the Selberg trace formula , etc.

The main novelty of Langlands' works, in addition to technical depth, consisted in hypotheses about a direct connection between the theory of automorphic forms and the theory of representations with number theory, in particular, about the correspondence between morphisms in these theories ( functoriality ).

For example, in Harish-Chandra's work, one can find a principle according to which what can be done for one semisimple (or reductive) Lie group should be done for everyone. Therefore, as soon as the role of some small-sized Lie groups, such as $\mathrm {GL} (2)$ ${\ displaystyle \ mathrm {GL} (2)}$ in the theory of modular forms, and with a retrospective look $\mathrm {GL} (1)$ ${\ displaystyle \ mathrm {GL} (1)}$ in class field theory , the path was open at least to the assumption of $\mathrm {GL} (n)$ ${\ displaystyle \ mathrm {GL} (n)}$ for the general case $n>2$ ${\ displaystyle n> 2}$ .

The idea of cusp form arose from points on modular curves , but also made sense, visible in spectral theory as a discrete spectrum , contrasting with a continuous spectrum from the Eisenstein series . It becomes much more technical for large Lie groups because parabolic subgroups are more numerous.

In all these approaches there was no shortage of technical methods, often inductive in nature and based on Levy decomposition among other issues, but the field was and remains very demanding ^[3] .

On the side of the modular forms were examples such as Hilbert modular forms, Siegel modular forms, and theta series .

Hypothesis Objects

There are a number of related Langlands hypotheses. There are many different groups in many different areas for which they can be stated, and for each area there are several different hypotheses ^[2] . Some versions of the Langlands hypothesis are vague or depend on objects such as Langlands groups whose existence is unproven, or on an L- group that has several nonequivalent definitions. Moreover, Langlands hypotheses have been developing since Langlands first stated them in 1967. .

There are various types of objects for which Langlands hypotheses can be formulated:

Representations of reductive groups over local fields (with different subcases corresponding to Archimedean local fields, p -adic local fields and completions of function fields )
Automorphic forms on reductive groups over local fields (with subcases corresponding to numerical fields or function fields).
End fields . Langlands did not initially consider this case, but his hypotheses have analogues for him.
More general fields, such as function fields over a complex number field.

Hypotheses

There are several different ways of presenting Langlands hypotheses that are closely related, but not obviously equivalent.

Reciprocity

The starting point of the program is Artin’s reciprocity law , which generalizes the quadratic reciprocity law . Artin’s reciprocity law applies to any Galois extension of an algebraic number field whose Galois group is Abelian ; he associates with the one-dimensional representations of this Galois group some L -functions and claims that these L -functions are identical to some Dirichlet L- series or more general series constructed according to Hecke's characters (that is, to some analogues of the Riemann zeta-function , for example, L - Hecke functions ). The exact correspondence between these various types of L- functions is Artin’s reciprocity law.

For non-Abelian Galois groups and their representations of dimension more than 1, one can also naturally define L-functions: Artin L-functions.

Langlands' insight was to find the correct generalization of Dirichlet L-functions, which would generalize Artin's formulation. Hecke previously associated Dirichlet L -functions with automorphic forms ( holomorphic functions on the upper half-plane $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ which satisfy some functional equations). Langlands then generalized them to automorphic caspidal representations , which are defined infinite-dimensional irreducible representations of a general linear group $\mathrm {GL} (n)$ ${\ displaystyle \ mathrm {GL} (n)}$ over the adele ring $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ . (This ring simultaneously tracks all recharge $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ , see p-adic numbers .)

Langlands linked automorphic L-functions to these automorphic representations and suggested that every Artin L- function arising from a finite-dimensional representation of a Galois group of a number field is equal to some L -function arising from an automorphic caspidal representation. This is known as his reciprocity hypothesis .

Roughly speaking, the reciprocity hypothesis gives a correspondence between automorphic representations of a reductive group and homomorphisms from the Langlands group to L-groups . There are many variations to this, partly because the definitions of the Langlands group and the L- group are not fixed.

It is expected that this will give a parametrization of L- packets of admissible irreducible representations of the reductive group over the local field. For example, over the field of real numbers, this correspondence is the Langlands classification of representations of real reductive groups. Over global fields, this correspondence should give a parametrization of automorphic forms.

Functionality

The functoriality hypothesis states that a suitable homomorphism of L- groups should give a correspondence between automorphic forms (in the global case) or representations (in the local case). Roughly speaking, the Langlands equivalence hypothesis is a special case of the functoriality hypothesis when one of the reductive groups is trivial.

Generalized Functionality

Langlands generalized the idea of functoriality: instead of a general linear group $\mathrm {GL} (n)$ ${\ displaystyle \ mathrm {GL} (n)}$ You can use other connected reductive groups . Moreover, having such a group $G$ ${\ displaystyle G}$ Langlands builds a dual band $^{L}G$ ${\ displaystyle ^ {L} G}$ and then for each automorphic caspidal representation $G$ ${\ displaystyle G}$ and any finite-dimensional representation $^{L}G$ ${\ displaystyle ^ {L} G}$ , it defines an L- function. One of his hypotheses states that these L -functions satisfy some functional equation that generalizes the functional equations of other known L- functions .

He then formulates a very general Principle of Functoriality . For two given reductive groups and a (good) morphism between the corresponding L- groups, the Functionality Principle connects their automorphic representations so that they are compatible with their L- functions. Many other existing hypotheses follow from this. This is the nature of the construction of the induced representation , which in the more traditional theory of automorphic forms was called “ uplift ”, known in special cases, and therefore covariant (whereas the limited representation is contravariant). Attempts to indicate a direct construction gave only some conditional results.

All these hypotheses can be formulated for more general fields instead of $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ : field of algebraic numbers (initial and most important case), local fields and function fields (finite extensions $\mathbb {F} _{p}(t)$ ${\ displaystyle \ mathbb {F} _ {p} (t)}$ - fields of rational functions over a finite field with $p$ ${\ displaystyle p}$ elements).

Geometric hypotheses

The so-called Langlands geometric program, proposed by Gerard Lomon following the ideas of Vladimir Drinfeld , arises from the geometric reformulation of the usual Langlands program. In simple cases, it binds $\ell$ ${\ displaystyle \ ell}$ -adic representations of the étale fundamental group of an algebraic curve with objects of a derived category $\ell$ ${\ displaystyle \ ell}$ -adic pencils on the modules of vector bundles over a curve.

Current Status

Langlands hypothesis for $\mathrm {GL} (1,K)$ ${\ displaystyle \ mathrm {GL} (1, K)}$ follow from (and are essentially equivalent to) class theory of fields .

Langlands proved Langlands conjectures for groups over Archimedean local fields $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ and $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ giving a classification of Langlands irreducible representations over these fields.

Lustig's classification of irreducible representations of groups of Lie type over finite fields can be considered as an analogue of the Langlands hypotheses for finite fields.

The proof of modularity of semistable elliptic curves over rational numbers given by Andrew Wiles can be considered as an example of the Langlands reciprocity hypothesis, since the main idea is to connect the Galois representations arising from elliptic curves with modular forms. Although Wiles’s results have been significantly generalized in many different directions, the complete Langlands hypothesis for $\mathrm {GL} (2,\mathbb {Q} )$ ${\ displaystyle \ mathrm {GL} (2, \ mathbb {Q})}$ remains unproven.

Laurent Luffforge proved Luffford's theorem - the Langlands conjecture for a general linear group $\mathrm {GL} (n,K)$ ${\ displaystyle \ mathrm {GL} (n, K)}$ for function fields $K$ ${\ displaystyle K}$ . This work continued earlier studies by Drinfeld, who proved the hypothesis for the case. $\mathrm {GL} (2,K)$ ${\ displaystyle \ mathrm {GL} (2, K)}$ .

Local Langlands hypotheses

Philip Kutsko in 1980 proved local Langlands conjectures for a general linear group $\mathrm {GL} (2,K)$ ${\ displaystyle \ mathrm {GL} (2, K)}$ over local fields.

Gerard Lomon , Mikhail Rapoport , Ulrich Stüler in 1993 proved the local Langlands hypotheses for a general linear group $\mathrm {GL} (n,K)$ ${\ displaystyle \ mathrm {GL} (n, K)}$ for local fields $K$ ${\ displaystyle K}$ positive characteristics. Their proof uses a global argument.

Richard Taylor , Michael Harris in 2001 proved local Langlands conjectures for a general linear group $\mathrm {GL} (n,K)$ ${\ displaystyle \ mathrm {GL} (n, K)}$ for local fields $K$ ${\ displaystyle K}$ characteristics 0. Guy Henniart in 2000 gave further evidence. Both proofs use a global argument. Peter Scholze in 2013 gave another proof.

Fundamental Lemma

In 2008, Ngo Bao Tyau proved: a fundamental lemma , which was originally proposed by Langlands in 1983 and required to prove some important hypotheses in the Langlands program ^[4] ^[5] .

Notes

↑ Math Quartet Joins Forces on Unified Theory (Neopr.) . Quanta (December 8, 2015).
↑ ¹ ² Frenkel, Edward (2015), Love and mathematics. The Heart of Hidden Reality , Peter, ISBN 978-5-496-01121-1
↑ “All this, as my dad put it, is a little heavy: here we have both Hitchin’s module spaces, mirror symmetry, A-branes, B-branes, automorphic bundles ... Trying to keep track of all the ingredients, you can easily get a headache ! Believe me, even among specialists, only a few can boast of understanding all aspects of this design ” ^[2]
↑ Ham Chau. Ngo Bao Chau, sommite mondiale des maths (Fr.) . Le Courrier du Vietnam (15 fevrier 2009).
↑ Langlands, Robert P. (1983), Les debuts d'une formule des traces stable , vol. 13, Publications Mathematiques de l'Universite Paris VII [Mathematical Publications of the University of Paris VII], Paris: Universite de Paris VII UER de Mathematiques , < http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/endoscopy. html # debuts >

Links

Arthur, James (2003), " The principle of functoriality ", American Mathematical Society. Bulletin. New Series T. 40 (1): 39–53, ISSN 0002-9904 , DOI 10.1090 / S0273-0979-02-00963-1
Bernstein, J. & Gelbart, S. (2003), An Introduction to the Langlands Program , Boston: Birkhauser, ISBN 3-7643-3211-5
- J. Bernstein, Art. Gelbart. Introduction to the Langlands program. - Moscow - Izhevsk, 2008.
Gelbart, Stephen (1984), " An elementary introduction to the Langlands program ", American Mathematical Society. Bulletin. New Series T. 10 (2): 177–219, ISSN 0002-9904 , DOI 10.1090 / S0273-0979-1984-15237-6
Frenkel, Edward (2005), "Lectures on the Langlands Program and Conformal Field Theory", arΧiv : hep-th / 0512172

Gelfand, IM (1963), "Automorphic functions and the theory of representations" , Proc. Internat. Congr. Mathematicians (Stockholm, 1962) , Djursholm: Inst. Mittag-Leffler, p. 74–85 , < http://mathunion.org/ICM/ICM1962.1/ > . Retrieved July 13, 2018. Archived July 17, 2011 on Wayback Machine
Harris, Michael & Taylor, Richard (2001), The geometry and cohomology of some simple Shimura varieties , vol. 151, Annals of Mathematics Studies, Princeton University Press , ISBN 978-0-691-09090-0 , < https://books.google.com/books?id=sigBbO69hvMC >
Henniart, Guy (2000), " Une preuve simple des conjectures de Langlands pour GL (n) sur un corps p-adique ", Inventiones Mathematicae T. 139 (2): 439–455, ISSN 0020-9910 , DOI 10.1007 / s002220050012
Kutzko, Philip (1980), " The Langlands Conjecture for Gl_2 of a Local Field ", Annals of Mathematics T. 112 (2): 381-412 , DOI 10.2307 / 1971151
Langlands, Robert (1967), Letter to Prof. Weil , < http://publications.ias.edu/rpl/section/21 >
Langlands, RP (1970), "Problems in the theory of automorphic forms" , Lectures in modern analysis and applications, III , vol. 170, Lecture Notes in Math, Berlin, New York: Springer-Verlag , p. 18–61, ISBN 978-3-540-05284-5 , doi : 10.1007 / BFb0079065 , < http://publications.ias.edu/rpl/section/21 >
Laumon, G .; Rapoport, M. & Stuhler, U. (1993), " D-elliptic sheaves and the Langlands correspondence ", Inventiones Mathematicae T. 113 (2): 217–338, ISSN 0020-9910 , DOI 10.1007 / BF01244308
Scholze, Peter (2013), " The Local Langlands Correspondence for GL (n) over p- adic fields ", Inventiones Mathematicae T. 192 (3): 663–715 , DOI 10.1007 / s00222-012-0420-5
Solomon Friedberg. What is ... the Langlands program? // Notices of the AMS . - 2018 .-- Vol. 65. - P. 663-665. - DOI : 10.1090 / noti1686 .
Vladimir Korolev. Connecting the incompatible (unopt.) . N + 1 (March 23, 2018). Date of treatment July 13, 2018.

Links

The work of robert langlands
Robert Langlands. Functionality and Reciprocity

[1] Math Quartet Joins Forces on Unified Theory (Neopr.) . Quanta (December 8, 2015).

[FrenkelLoveMath-2] ¹ ² Frenkel, Edward (2015), Love and mathematics. The Heart of Hidden Reality , Peter, ISBN 978-5-496-01121-1

[3] “All this, as my dad put it, is a little heavy: here we have both Hitchin’s module spaces, mirror symmetry, A-branes, B-branes, automorphic bundles ... Trying to keep track of all the ingredients, you can easily get a headache ! Believe me, even among specialists, only a few can boast of understanding all aspects of this design ” ^[2]

[4] Ham Chau. Ngo Bao Chau, sommite mondiale des maths (Fr.) . Le Courrier du Vietnam (15 fevrier 2009).

[5] Langlands, Robert P. (1983), Les debuts d'une formule des traces stable , vol. 13, Publications Mathematiques de l'Universite Paris VII [Mathematical Publications of the University of Paris VII], Paris: Universite de Paris VII UER de Mathematiques , < http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/endoscopy. html # debuts >

Langlands Program

Content

Context

Hypothesis Objects

Hypotheses

Reciprocity

Functionality

Generalized Functionality

Geometric hypotheses

Current Status

Local Langlands hypotheses

Fundamental Lemma

Notes

Links

Links

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