Misplaced Pages

#179820

71-520: The theory of L -functions has become a very substantial, and still largely conjectural , part of contemporary analytic number theory . In it, broad generalisations of the Riemann zeta function and the L -series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the Euler product formula there

142-445: A {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} can satisfy the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} for any integer value of n {\displaystyle n} greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in

213-422: A number field to its own algebraic structure . The meaning of such a construction is nuanced, but its specific solutions and generalizations are useful. The consequence for proof of existence to such theoretical objects implies an analytical method for constructing the categoric mapping of fundamental structures for virtually any number field . As an analogue to the possible exact distribution of primes ,

284-434: A theorem , proven in 1995 by Andrew Wiles ), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down

355-572: A theorem . Many important theorems were once conjectures, such as the Geometrization theorem (which resolved the Poincaré conjecture ), Fermat's Last Theorem , and others. Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf. the Pólya conjecture and Euler's sum of powers conjecture ). In the case of the latter, the first counterexample found for

426-404: 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 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

497-427: A common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze . This conjecture is now known to be false. The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion . The manifold version is true in dimensions m ≤ 3 . The cases m = 2 and 3 were proved by Tibor Radó and Edwin E. Moise in

568-471: A finite-dimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation. This is known as his reciprocity conjecture . Roughly speaking, this conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L -group . This offers numerous variations, in part because

639-683: A natural way: Artin L -functions . Langlands' insight was to find the proper generalization of Dirichlet L -functions , which would allow the formulation of Artin's statement in Langland's more general setting. Hecke had earlier related Dirichlet L -functions with automorphic forms ( holomorphic functions on the upper half plane of the complex number plane C {\displaystyle \mathbb {C} } that satisfy certain functional equations ). Langlands then generalized these to automorphic cuspidal representations , which are certain infinite dimensional irreducible representations of

710-478: A parameterization of automorphic forms. 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. Langlands generalized

781-440: A proof that the conjecture is true—because the conjecture might be false but with a very large minimal counterexample. Nevertheless, mathematicians often regard a conjecture as strongly supported by evidence even though not yet proved. That evidence may be of various kinds, such as verification of consequences of it or strong interconnections with known results. A conjecture is considered proven only when it has been shown that it

SECTION 10

#1732779981180

852-523: A smaller counter-example). Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this with hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that

923-427: A space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time. After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv . The proof followed on from

994-638: 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 ) constructed from Hecke characters . 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, L -functions can be defined in

1065-408: Is a prime and F p ( t ) is the field of rational functions over the finite field with p elements). The geometric Langlands program, suggested by Gérard Laumon following ideas of Vladimir Drinfeld , arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates l -adic representations of

1136-411: Is a deep connection between L -functions and the theory of prime numbers . The mathematical field that studies L -functions is sometimes called analytic theory of L -functions . We distinguish at the outset between the L -series , an infinite series representation (for example the Dirichlet series for the Riemann zeta function ), and the L -function , the function in the complex plane that

1207-516: Is a major unsolved problem in computer science . Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer; it is widely conjectured that the answer is no. It was essentially first mentioned in a 1956 letter written by Kurt Gödel to John von Neumann . Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time. The precise statement of

1278-485: Is correct. The Poincaré conjecture, before being proven, was one of the most important open questions in topology . In mathematics, the Riemann hypothesis , proposed by Bernhard Riemann  ( 1859 ), is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields . The Riemann hypothesis implies results about

1349-402: Is found over zeros covering at least fifteen orders of magnitude for the Riemann zeta function , and also for the zeros of other L -functions of different orders and conductors. One of the influential examples, both for the history of the more general L -functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of

1420-441: Is its analytic continuation . The general constructions start with an L -series, defined first as a Dirichlet series , and then by an expansion as an Euler product indexed by prime numbers. Estimates are required to prove that this converges in some right half-plane of the complex numbers . Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles ). It

1491-410: Is logically impossible for it to be false. There are various methods of doing so; see methods of mathematical proof for more details. One method of proof, applicable when there are only a finite number of cases that could lead to counterexamples, is known as " brute force ": in this approach, all possible cases are considered and shown not to give counterexamples. In some occasions, the number of cases

SECTION 20

#1732779981180

1562-408: Is quite large, in which case a brute-force proof may require as a practical matter the use of a computer algorithm to check all the cases. For example, the validity of the 1976 and 1997 brute-force proofs of the four color theorem by computer was initially doubted, but was eventually confirmed in 2005 by theorem-proving software. When a conjecture has been proven , it is no longer a conjecture but

1633-505: Is the biggest project in mathematical research. It was described by Edward Frenkel as " grand unified theory of mathematics." A non-specialist description: the construct of a generalised and somewhat unified framework to characterise the structures that underpin numbers and their abstractions , and thus the invariants which base them, all through analytical methods. The Langlands program consists of theoretical abstractions, which challenge even specialist mathematicians. Basically,

1704-505: Is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner (much as Euclid 's parallel postulate can be taken either as true or false in an axiomatic system for geometry). In this case, if a proof uses this statement, researchers will often look for a new proof that does not require the hypothesis (in the same way that it is desirable that statements in Euclidean geometry be proved using only

1775-416: Is this (conjectural) meromorphic continuation to the complex plane which is called an L -function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the L -function at points where the series representation does not converge. The general term L -function here includes many known types of zeta functions. The Selberg class is an attempt to capture

1846-467: Is unproven, or on the L -group that has several non-equivalent definitions. Objects for which Langlands conjectures can be stated: The conjectures can be stated variously in ways that are closely related but not obviously equivalent. The starting point of the program was Emil Artin 's reciprocity law , which generalizes quadratic reciprocity . The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group

1917-410: The Riemann hypothesis is a conjecture from number theory that — amongst other things — makes predictions about the distribution of prime numbers . Few number theorists doubt that the Riemann hypothesis is true. In fact, in anticipation of its eventual proof, some have even proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called conditional proofs :

1988-462: The fundamental lemma of the project links the generalized fundamental representation of a finite field with its group extension to the automorphic forms under which it is invariant . This is accomplished through abstraction to higher dimensional integration , by an equivalence to a certain analytical group as an absolute extension of its algebra . This allows an analytical functional construction of powerful invariance transformations for

2059-425: The general linear group GL( n ) over the adele ring of Q {\displaystyle \mathbb {Q} } (the rational numbers ). (This ring tracks all the completions of Q , {\displaystyle \mathbb {Q} ,} see p -adic numbers .) Langlands attached automorphic L -functions to these automorphic representations, and conjectured that every Artin L -function arising from

2130-438: The invariance within structures of number fields. Additionally, some connections between the Langlands program and M theory have been posited, as their dualities connect in nontrivial ways, providing potential exact solutions in superstring theory (as was similarly done in group theory through monstrous moonshine ). Simply put, the Langlands project implies a deep and powerful framework of solutions, which touches

2201-418: The local Langlands conjectures for the general linear group GL(2, K ) over local fields. Gérard Laumon , Michael Rapoport , and Ulrich Stuhler  ( 1993 ) 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. Michael Harris and Richard Taylor  ( 2001 ) proved

L-function - Misplaced Pages Continue

2272-499: The étale fundamental group of an algebraic curve to objects of the derived category of l -adic sheaves on the moduli stack of vector bundles over the curve. A 9-person collaborative project led by Dennis Gaitsgory announced a proof of the (categorical, unramified) geometric Langlands conjecture leveraging Hecke eigensheaf as part of the proof. The Langlands conjectures for GL(1, K ) follow from (and are essentially equivalent to) class field theory . Langlands proved

2343-588: The 1920s and 1950s, respectively. In mathematics , the Weil conjectures were some highly influential proposals by André Weil  ( 1949 ) on the generating functions (known as local zeta-functions ) derived from counting the number of points on algebraic varieties over finite fields . A variety V over a finite field with q elements has a finite number of rational points , as well as points over every finite field with q elements containing that field. The generating function has coefficients derived from

2414-402: The 1960s. It applies to an elliptic curve E , and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or another global field ): i.e. the number of free generators of its group of rational points. Much previous work in the area began to be unified around a better knowledge of L -functions. This was something like a paradigm example of

2485-496: The 20th century. It is among the most notable theorems in the history of mathematics , and prior to its proof it was in the Guinness Book of World Records for "most difficult mathematical problems". In mathematics , the four color theorem , or the four color map theorem, states that given any separation of a plane into contiguous regions, producing a figure called a map , no more than four colors are required to color

2556-512: The 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence : if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it. Originally conjectured by Henri Poincaré in 1904, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold ). The Poincaré conjecture claims that if such

2627-623: The Langlands conjectures for finite fields. Andrew Wiles ' 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 GL ( 2 , Q ) {\displaystyle {\text{GL}}(2,\mathbb {Q} )} remains unproved. In 1998, Laurent Lafforgue proved Lafforgue's theorem verifying

2698-512: The Langlands conjectures for groups over the archimedean local fields R {\displaystyle \mathbb {R} } (the real numbers ) and C {\displaystyle \mathbb {C} } (the complex numbers ) by giving the Langlands classification 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

2769-443: 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 established the global Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields. Philip Kutzko  ( 1980 ) proved

2840-403: The Langlands program allows a potential general tool for the resolution of invariance at the level of generalized algebraic structures . This in turn permits a somewhat unified analysis of arithmetic objects through their automorphic functions . The Langlands concept allows a general analysis of structuring number abstractions. This description is at once a reduction and over-generalization of

2911-530: The P=NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$ 1,000,000 prize for the first correct solution. Karl Popper pioneered the use of

L-function - Misplaced Pages Continue

2982-523: The Riemann zeta function connects through its values at positive even integers (and negative odd integers) to the Bernoulli numbers , one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for p -adic L -functions , which describe certain Galois modules . The statistics of the zero distributions are of interest because of their connection to problems like

3053-423: The analytic sense: there should be some input from analysis, which meant automorphic analysis. The general case now unifies at a conceptual level a number of different research programs. Conjectural 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 or Fermat's conjecture (now

3124-422: The axioms of neutral geometry, i.e. without the parallel postulate). The one major exception to this in practice is the axiom of choice , as the majority of researchers usually do not worry whether a result requires it—unless they are studying this axiom in particular. Sometimes, a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example,

3195-497: The conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture , which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 10 (1.2 trillion). However, the failure to find a counterexample after extensive search does not constitute

3266-429: The conjectures assumed appear in the hypotheses of the theorem, for the time being. These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type. In number theory , Fermat's Last Theorem (sometimes called Fermat's conjecture , especially in older texts) states that no three positive integers

3337-406: The core properties of L -functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions. One can list characteristics of known examples of L -functions that one would wish to see generalized: Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since

3408-405: The definitions of Langlands group and L -group are not fixed. Over local fields 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 of representations of real reductive groups. Over global fields , it should give

3479-658: The distribution of prime numbers . Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics . The Riemann hypothesis, along with the Goldbach conjecture , is part of Hilbert's eighth problem in David Hilbert 's list of 23 unsolved problems ; it is also one of the Clay Mathematics Institute Millennium Prize Problems . The P versus NP problem

3550-507: The field remained demanding. From the perspective of modular forms, examples such as Hilbert modular forms , Siegel modular forms , and theta-series had been developed. The conjectures have evolved since Langlands first stated them. Langlands conjectures apply across many different groups over many different fields for which they can be stated, and each field offers several versions of the conjectures. Some versions are vague, or depend on objects such as Langlands groups , whose existence

3621-440: The first statement of the four color theorem in 1852. The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken . It was the first major theorem to be proved using a computer . Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make

SECTION 50

#1732779981180

3692-416: The functional equation by Grothendieck (1965) , and the analogue of the Riemann hypothesis was proved by Deligne (1974) . In mathematics , the Poincaré conjecture is a theorem about the characterization of the 3-sphere , which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states that: Every simply connected , closed 3- manifold is homeomorphic to

3763-414: The generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with random matrix theory and quantum chaos are also of interest. The fractal structure of the distributions has been studied using rescaled range analysis. The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension

3834-437: The idea of 'Functoriality' between abstract algebraic representations of number fields and their analytical prime constructions results in powerful functional tools allowing an exact quantification of prime distributions . This, in turn, yields the capacity for classification of diophantine equations and further abstractions of algebraic functions . Furthermore, if the reciprocity of such generalized algebras for

3905-423: The idea of functoriality: instead of using the general linear group GL( n ), other connected reductive groups can be used. Furthermore, given such a group G , Langlands constructs the Langlands dual group G , and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of G , he defines an L -function. One of his conjectures states that these L -functions satisfy

3976-403: The local Langlands conjectures for the general linear group GL( n , K ) for characteristic 0 local fields K . Guy Henniart  ( 2000 ) gave another proof. Both proofs use a global argument. Peter Scholze  ( 2013 ) gave another proof. In 2008, Ngô Bảo Châu proved the " fundamental lemma ", which was originally conjectured by Langlands and Shelstad in 1983 and being required in

4047-408: The margin of a copy of Arithmetica , where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles , and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century, and the proof of the modularity theorem in

4118-426: The n=4 case involved numbers in the millions, although it has been subsequently found that the minimal counterexample is actually smaller. Not every conjecture ends up being proven true or false. The continuum hypothesis , which tries to ascertain the relative cardinality of certain infinite sets , was eventually shown to be independent from the generally accepted set of Zermelo–Fraenkel axioms of set theory. It

4189-500: The nascent theory of L -functions. This development preceded the Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin L -functions , which, like Hecke L -functions , were defined several decades earlier, and to L -functions attached to general automorphic representations . Gradually it became clearer in what sense the construction of Hasse–Weil zeta functions might be made to work to provide valid L -functions, in

4260-418: The numbers N k of points over the (essentially unique) field with q elements. Weil conjectured that such zeta-functions should be rational functions , should satisfy a form of functional equation , and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis . The rationality was proved by Dwork (1960) ,

4331-673: The other conjectures presented so far. It is of the nature of an induced representation construction—what in the more traditional theory of automorphic forms had been called a ' lifting ', known in special cases, and so is covariant (whereas a restricted representation 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 Q {\displaystyle \mathbb {Q} } : algebraic number fields (the original and most important case), local fields , and function fields (finite extensions of F p ( t ) where p

SECTION 60

#1732779981180

4402-446: 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 . Such proofs would be expected to utilize abstract solutions in objects of generalized analytical series , each of which relates to

4473-537: The problem in his lectures as early as 1840. The conjecture was first proposed on October 23, 1852 when Francis Guthrie , while trying to color the map of counties of England, noticed that only four different colors were needed. The five color theorem , which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century; however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since

4544-511: The program of Richard S. Hamilton to use the Ricci flow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method "converged" in three dimensions. Perelman completed this portion of the proof. Several teams of mathematicians have verified that Perelman's proof

4615-447: The program's proper theorems, although these mathematical analogues provide its basis. The Langlands program is built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and Gelfand  ( 1963 ), the work and Harish-Chandra's approach on semisimple Lie groups , and in technical terms the trace formula of Selberg and others. What was new in Langlands' work, besides technical depth,

4686-437: The proof of some important conjectures in the Langlands program. 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 and generalizations of algebraic geometry ,

4757-502: The regions of the map—so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not. Möbius mentioned

4828-536: The term "conjecture" in scientific philosophy . Conjecture is related to hypothesis , which in science refers to a testable conjecture. Langlands program In mathematics , the Langlands program is a set of conjectures about connections between number theory and geometry . It was proposed by Robert Langlands  ( 1967 , 1970 ). It seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles . It

4899-424: The theorem is therefore true. Initially, their proof was not accepted by mathematicians at all because the computer-assisted proof was infeasible for a human to check by hand. However, the proof has since then gained wider acceptance, although doubts still remain. The Hauptvermutung (German for main conjecture) of geometric topology is the conjecture that any two triangulations of a triangulable space have

4970-537: The way was open 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 as " discrete spectrum ", contrasted with the " continuous spectrum " from Eisenstein series . It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous. In all these approaches technical methods were available, often inductive in nature and based on Levi decompositions amongst other matters, but

5041-453: Was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-called functoriality ). Harish-Chandra's work exploited the principle that what can be done for one semisimple (or reductive) Lie group , can 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 ,

#179820