In
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
and
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, the Langlands program is a web of far-reaching and influential
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
s about connections between
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. Proposed by , it seeks to relate
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
s in
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
to
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s and
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
s over
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s and
adele
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a reco ...
s. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by
Edward Frenkel
Edward Vladimirovich Frenkel (; born May 2, 1968) is a Russian-American mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley, a member ...
as "a kind of
grand unified theory
A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this ...
of mathematics."
The Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. To oversimplify, the
fundamental lemma of the project posits a direct connection between the generalized
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group
or Lie algebra whose highest weight is a fundamental weight. For example, the defini ...
of a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with its
group extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence
:1\to N\;\overs ...
to the
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s under which it is
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
. This is accomplished through abstraction to
higher dimensional integration, by an equivalence to a certain
analytical group as an
absolute extension of its
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
. Consequently, this allows an
analytical functional construction of powerful
invariance transformations for a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
to its own
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
.
The meaning of such a construction is nuanced, but its specific solutions and generalizations are very powerful. The consequence for proof of existence to such theoretical objects implies an
analytical method Analytical technique is a method used to determine a chemical or physical property of a chemical substance, chemical element, or mixture. There is a wide variety of techniques used for analysis, from simple weighing to advanced techniques using high ...
in constructing the
categoric mapping of
fundamental structure
In Schenkerian analysis, the fundamental structure (german: Ursatz) describes the structure of a tonal work as it occurs at the most remote (or "background") level and in the most abstract form. A basic elaboration of the tonic triad, it consist ...
s for virtually any
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
. As an analogue to the possible
exact distribution of primes, the Langlands program allows a potential
general tool for the resolution of invariance at the level of generalized
algebraic structures
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
. This in turn permits a somewhat unified analysis of
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
objects through their
automorphic functions. Simply put, the Langlands philosophy allows a general analysis of structuring the abstractions of numbers. Naturally, this description is at once a reduction and over-generalization of the program's proper theorems, but these mathematical analogues provide the basis of its conceptualization.
Background
In a very broad context, the program built on existing ideas: the ''
philosophy of cusp forms'' formulated a few years earlier by
Harish-Chandra
Harish-Chandra Fellow of the Royal Society, FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
...
and , the work and approach of Harish-Chandra on
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, i ...
s, and in technical terms the
trace formula of
Selberg and others.
What initially was very new in Langlands' work, besides technical depth, was the proposed direct connection to number theory, together with the rich organisational structure hypothesised (so-called ''
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
iality'').
For example, in the work of Harish-Chandra one finds the principle that what can be done for one
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
(or reductive)
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
, should be done for all. Therefore, once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, and with hindsight GL(1) in
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
, the way was open at least to speculation about GL(''n'') for general ''n'' > 2.
The ''cusp form'' idea came out of the cusps on
modular curves but also had a meaning visible in
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
as "
discrete spectrum
A observable, physical quantity is said to have a discrete spectrum if it takes only distinct values, with gaps between one value and the next.
The classical example of discrete spectrum (for which the term was first used) is the characterist ...
", contrasted with the "
continuous spectrum" from
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
. It becomes much more technical for bigger Lie groups, because the
parabolic subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
s are more numerous.
In all these approaches there was no shortage of technical methods, often inductive in nature and based on
Levi decomposition
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semi ...
s amongst other matters, but the field wasand isvery demanding.
And on the side of modular forms, there were examples such as
Hilbert modular forms,
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...
s, and
theta-series.
Objects
There are a number of related Langlands conjectures. There are many different groups over many different fields for which they can be stated, and for each field there are several different versions of the conjectures. Some versions of the Langlands conjectures are vague, or depend on objects such as the
Langlands group
In mathematics, the Langlands group is a conjectural group ''L'F'' attached to each local or global field ''F'', that satisfies properties similar to those of the Weil group. It was given that name by Robert Kottwitz. In Kottwitz's formulatio ...
s, whose existence is unproven, or on the ''L''-group that has several inequivalent definitions. Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1967.
There are different types of objects for which the Langlands conjectures can be stated:
*Representations of
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
s 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 global fields (with subcases corresponding to number fields or function fields).
*Finite fields. Langlands did not originally consider this case, but his conjectures have analogues for it.
*More general fields, such as function fields over the complex numbers.
Conjectures
There are several different ways of stating the Langlands conjectures, which are closely related but not obviously equivalent.
Reciprocity
The starting point of the program may be seen as
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
's
reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irr ...
, which generalizes
quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
. The
Artin reciprocity law The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...
applies to a
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
of an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
whose
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
is
abelian; it assigns
''L''-functions to the one-dimensional representations of this Galois group, and states that these ''L''-functions are identical to certain
Dirichlet ''L''-series or more general series (that is, certain analogues of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
) constructed from
Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of
''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ce ...
s. The precise correspondence between these different kinds of ''L''-functions constitutes Artin's reciprocity law.
For non-abelian Galois groups and higher-dimensional representations of them, one can still define ''L''-functions in a natural way:
Artin ''L''-functions.
The insight of Langlands was to find the proper generalization of Dirichlet ''L''-functions, which would allow the formulation of Artin's statement in this more general setting.
Hecke had earlier related Dirichlet ''L''-functions with
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s (
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s on the upper half plane of
(the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s) that satisfy certain functional equations). Langlands then generalized these to
automorphic cuspidal representation
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...
s, which are certain infinite dimensional irreducible representations of the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(''n'') over the
adele ring
Adele Laurie Blue Adkins (, ; born 5 May 1988), professionally known by the mononym Adele, is an English singer and songwriter. After graduating in arts from the BRIT School in 2006, Adele signed a reco ...
of
(the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s). (This ring simultaneously keeps track of all the completions of
see
''p''-adic numbers.)
Langlands attached
automorphic ''L''-functions to these automorphic representations, and conjectured that every Artin ''L''-function arising from a finite-dimensional representation of the Galois group of a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
is equal to one arising from an automorphic cuspidal representation. This is known as his "
reciprocity conjecture".
Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a
Langlands group
In mathematics, the Langlands group is a conjectural group ''L'F'' attached to each local or global field ''F'', that satisfies properties similar to those of the Weil group. It was given that name by Robert Kottwitz. In Kottwitz's formulatio ...
to an
''L''-group. There are numerous variations of this, in part because the definitions of Langlands group and ''L''-group are not fixed.
Over
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s this is expected to give a parameterization of
''L''-packets of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers, this correspondence is the
Langlands classification In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group ''G'', suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One ...
of representations of real reductive groups. Over
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
s, it should give a parameterization of automorphic forms.
Functoriality
The functoriality conjecture states that a suitable homomorphism of ''L''-groups is expected to give a correspondence between automorphic forms (in the global case) or representations (in the local case). Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.
Generalized functoriality
Langlands generalized the idea of functoriality: instead of using the general linear group GL(''n''), other connected
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
s can be used. Furthermore, given such a group ''G'', Langlands constructs the
Langlands dual
In representation theory, a branch of mathematics, the Langlands dual ''L'G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a fie ...
group ''
LG'', and then, for every automorphic cuspidal representation of ''G'' and every finite-dimensional representation of ''
LG'', he defines an ''L''-function. One of his conjectures states that these ''L''-functions satisfy a certain functional equation generalizing those of other known ''L''-functions.
He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved)
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
between their corresponding ''L''-groups, this conjecture relates their automorphic representations in a way that is compatible with their ''L''-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an
induced representation
In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represent ...
construction—what in the more traditional theory of
automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s had been called a '
lifting', known in special cases, and so is covariant (whereas a
restricted representation In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to under ...
is contravariant). Attempts to specify a direct construction have only produced some conditional results.
All these conjectures can be formulated for more general fields in place of
:
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s (the original and most important case),
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s, and function fields (finite
extensions
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
of F
''p''(''t'') where ''p'' is a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
and F
''p''(''t'') is the field of rational functions over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with ''p'' elements).
Geometric conjectures
The so-called geometric Langlands program, suggested by
Gérard Laumon following ideas of
Vladimir Drinfeld
Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowne ...
, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates -adic representations of the
étale fundamental group The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group ''π''1(''X' ...
of an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
to objects of the
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
of -adic sheaves on the
moduli stack
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such space ...
of
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s over the curve.
Current status
The Langlands conjectures for GL(1, ''K'') follow from (and are essentially equivalent to)
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
.
Langlands proved the Langlands conjectures for groups over the archimedean local fields
(the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s) and
by giving the
Langlands classification In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group ''G'', suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One ...
of their irreducible representations.
Lusztig's classification of the irreducible representations of groups of Lie type over finite fields can be considered an analogue of the Langlands conjectures for finite fields.
Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...
' proof of modularity of semistable elliptic curves over rationals can be viewed as an instance of the Langlands reciprocity conjecture, since the main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture for
remains unproved.
In 1998,
Laurent Lafforgue
Laurent Lafforgue (; born 6 November 1966) is a French mathematician. He has made outstanding contributions to Langlands' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism ...
proved
Lafforgue's theorem verifying the Langlands conjectures for the general linear group GL(''n'', ''K'') for function fields ''K''. This work continued earlier investigations by Drinfeld, who proved the case GL(2, ''K'') in the 1980s.
In 2018,
Vincent Lafforgue
Vincent Lafforgue (born 20 January 1974) is a French mathematician who is active in algebraic geometry, especially in the Langlands program, and a CNRS " Directeur de Recherches" at the Institute Fourier in Grenoble. He is the younger brother o ...
established the global Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields.
Local Langlands conjectures
proved the
local Langlands conjectures In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ''G'' over a local field ''F'', and representation ...
for the general linear group GL(2, ''K'') over local fields.
proved the local Langlands conjectures for the general linear group GL(''n'', ''K'') for positive characteristic local fields ''K''. Their proof uses a global argument.
proved the local Langlands conjectures for the general linear group GL(''n'', ''K'') for characteristic 0 local fields ''K''. gave another proof. Both proofs use a global argument. gave another proof.
Fundamental lemma
In 2008,
Ngô Bảo Châu
Ngô Bảo Châu (, born June 28, 1972) is a Vietnamese-French mathematician at the University of Chicago, best known for proving the fundamental lemma for automorphic forms (proposed by Robert Langlands and Diana Shelstad). He is the first Vie ...
proved the "
fundamental lemma", which was originally conjectured by Langlands and Shelstad in 1983 and being required in the proof of some important conjectures in the Langlands program.
Implications
To a lay reader or even nonspecialist mathematician, abstractions within the Langlands program can be somewhat impenetrable. However, there are some strong and clear implications for proof or disproof of the fundamental Langlands conjectures.
As the program posits a powerful connection between
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
and generalizations of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the idea of
'functoriality' between abstract algebraic
representations
''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s and their analytical
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
constructions results in powerful
functional tools allowing an exact quantification of
prime distributions. This, in turn, yields the capacity for classification of
diophantine equations
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
and further abstractions of
algebraic functions.
Furthermore, if the
reciprocity of such generalized
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
for the posited objects exists, and if their
analytical functions can be shown to be well-defined, some very deep results in mathematics could be within reach of proof. Examples include:
rational solutions of elliptic curves,
topological construction of algebraic varieties, and the famous
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
. Such proofs would be expected to utilize abstract solutions in objects of
generalized analytical series, each of which relates to the
invariance within
structures
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
of number fields.
Additionally, some connections between the Langlands program and
M theory have been posited, as their
dualities connect in
nontrivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...
ways, providing potential exact solutions in
superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings.
'Superstring theory' is a shorthand for supersymmetric string theor ...
(as was similarly done in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
through
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
).
Simply put, the Langlands project implies a deep and powerful framework of solutions, which touches the most fundamental areas of mathematics, through high-order generalizations in exact solutions of algebraic equations, with analytical functions, as embedded in geometric forms. It allows a unification of many distant mathematical fields into a formalism of powerful
analytical methods.
See also
*
Jacquet–Langlands correspondence
*
Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
Notes
References
*
*
*
*
*
*
*
*
*
*
*
*
External links
The work of Robert Langlands
{{L-functions-footer
Zeta and L-functions
Representation theory of Lie groups
Automorphic forms
Conjectures
History of mathematics