25:
1033:' 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
397:
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
243:. 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.
1350:
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
1473:
The
Langlands Program is now a vast subject. There is a large community of people working on it in different fields: number theory, harmonic analysis, geometry, representation theory, mathematical physics. Although they work with very different objects, they are all observing similar
730:-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.
1237:
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.
1535:
Arinkin, D.; Beraldo, D.; Campbell, J.; Chen, L.; Faergeman, J.; Gaitsgory, D.; Lin, K.; Raskin, S.; Rozenblyum, N. (May 2024). "Proof of the geometric
Langlands conjecture II: Kac-Moody localization and the FLE".
1440:-branes, automorphic sheaves... One can get a headache just trying to keep track of them all. Believe me, even among specialists, very few people know the nuts and bolts of all elements of this construction.
307:
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
207:
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
1070:
324:, 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
1027:
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.
1106:
established the global
Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields.
646:
1014:
988:
832:
617:
575:
152:
The
Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. To oversimplify, the
910:, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates
1229:", which was originally conjectured by Langlands and Shelstad in 1983 and being required in the proof of some important conjectures in the Langlands program.
2548:
2280:
681:
Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a
710:
of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers, this correspondence is the
2477:
246:
In short, a simplified description for this theory to a non-specialist would be: the construct of a generalised and somewhat unified framework, to
406:-group that has several inequivalent definitions. Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1967.
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1941:
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There are several different ways of stating the
Langlands conjectures, which are closely related but not obviously equivalent.
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2431:
1172:
1168:
356:
352:
424:
Automorphic forms on reductive groups over global fields (with subcases corresponding to number fields or function fields).
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can be shown to be well-defined, some very deep results in mathematics could be within reach of proof. Examples include:
141:. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by
793:-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an
1389:
781:
He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved)
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39:
33:
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50:
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In all these approaches there was no shortage of technical methods, often inductive in nature and based on
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is contravariant). Attempts to specify a direct construction have only produced some conditional results.
794:
789:-groups, this conjecture relates their automorphic representations in a way that is compatible with their
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468:
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Finite fields. Langlands did not originally consider this case, but his conjectures have analogues for it.
317:
263:
236:
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2310:
2030:
1514:; Raskin, Sam (May 2024). "Proof of the geometric Langlands conjecture I: construction of the functor".
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193:
181:
82:
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Milne, James (2015-09-02). "The Riemann Hypothesis over Finite Fields: From Weil to the Present Day".
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1977:
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All this stuff, as my dad put it, is quite heavy: we've got Hitchin moduli spaces, mirror symmetry,
1344:
1278:
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For non-abelian Galois groups and higher-dimensional representations of them, one can still define
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amongst other matters, but the field was – and is – very demanding.
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2380:
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2009:
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has announced a proof of the (categorical, unramified) geometric Langlands conjecture leveraging
802:
491:
371:
325:
316:
For example, in the work of Harish-Chandra one finds the principle that what can be done for one
228:
224:
692:. There are numerous variations of this, in part because the definitions of Langlands group and
1665:
La paramétrisation de Langlands globale sur les corps des fonctions (d'après Vincent Lafforgue)
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2177:
2141:
2089:
1993:
1937:
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Langlands generalized the idea of functoriality: instead of using the general linear group GL(
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2005:
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1919:
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is equal to one arising from an automorphic cuspidal representation. This is known as his "
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2001:
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1981:
968:
Langlands proved the Langlands conjectures for groups over the archimedean local fields
409:
There are different types of objects for which the Langlands conjectures can be stated:
1888:
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1414:
1409:
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751:
553:
536:, which would allow the formulation of Artin's statement in this more general setting.
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over local fields (with different subcases corresponding to archimedean local fields,
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2013:
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98:
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1095:. This work continued earlier investigations by Drinfeld, who proved the case GL(2,
670:-function arising from a finite-dimensional representation of the Galois group of a
2517:
2512:
1340:
1258:
1254:
1192:
1030:
881:
864:
774:-functions satisfy a certain functional equation generalizing those of other known
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to the one-dimensional representations of this Galois group, and states that these
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134:
2258:
1573:"Shtukas for reductive groups and Langlands correspondence for function fields"
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2232:
1958:
Henniart, Guy (2000), "Une preuve simple des conjectures de Langlands pour GL(
1708:
1671:. Séminaire Bourbaki 68ème année, 2015–2016, no. 1110, Janvier 2016.
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94:
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1997:
1873:
1809:
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unification of many distant mathematical fields into a formalism of powerful
1308:. Such proofs would be expected to utilize abstract solutions in objects of
812:
All these conjectures can be formulated for more general fields in place of
585:, which are certain infinite dimensional irreducible representations of the
321:
1989:
1324:
1179:) proved the local Langlands conjectures for the general linear group GL(
1153:) proved the local Langlands conjectures for the general linear group GL(
782:
704:
102:
1891:(2005). "Lectures on the Langlands Program and Conformal Field Theory".
2137:
2085:
2051:
1897:
1640:
666:
to these automorphic representations, and conjectured that every Artin
309:
1907:
430:
More general fields, such as function fields over the complex numbers.
1615:"Chtoucas pour les groupes réductifs et paramétrisation de Langlands"
2043:
1083:
verifying the Langlands conjectures for the general linear group GL(
726:
The functoriality conjecture states that a suitable homomorphism of
363:. It becomes much more technical for bigger Lie groups, because the
2253:
1767:
1585:
1542:
1520:
623:). (This ring simultaneously keeps track of all the completions of
2215:
2191:
Scholze, Peter (2013), "The Local Langlands Correspondence for GL(
1699:
1631:
528:
The insight of Langlands was to find the proper generalization of
274:
In a very broad context, the program built on existing ideas: the
156:
of the project posits a direct connection between the generalized
1323:
Additionally, some connections between the Langlands program and
2262:
1850:(1984), "An elementary introduction to the Langlands program",
506:. The precise correspondence between these different kinds of
377:
And on the side of modular forms, there were examples such as
18:
16:
Far-reaching conjectures connecting number theory and geometry
758:, and then, for every automorphic cuspidal representation of
231:
for the resolution of invariance at the level of generalized
1929:
The geometry and cohomology of some simple Shimura varieties
1273:. This, in turn, yields the capacity for classification of
2076:, Lecture Notes in Math, vol. 170, Berlin, New York:
805:', known in special cases, and so is covariant (whereas a
718:, it should give a parameterization of automorphic forms.
498:
or more general series (that is, certain analogues of the
1908:"Automorphic functions and the theory of representations"
1685:(2010). "Le lemme fondamental pour les algèbres de Lie".
1558:"Monumental Proof Settles Geometric Langlands Conjecture"
1199:) gave another proof. Both proofs use a global argument.
1786:
Arthur, James (2003), "The principle of functoriality",
1912:
Proc. Internat. Congr. Mathematicians (Stockholm, 1962)
421:-adic local fields, and completions of function fields)
235:. This in turn permits a somewhat unified analysis of
2114:-elliptic sheaves and the Langlands correspondence",
1150:
1039:
1000:
974:
818:
629:
603:
561:
1241:
As the program posits a powerful connection between
797:
construction—what in the more traditional theory of
770:-function. One of his conjectures states that these
328:, the way was open at least to speculation about GL(
2491:
2455:
2419:
2393:
2350:
2303:
1914:, Djursholm: Inst. Mittag-Leffler, pp. 74–85,
1413:
1064:
1008:
982:
961:) follow from (and are essentially equivalent to)
826:
714:of representations of real reductive groups. Over
640:
611:
569:
2074:Lectures in modern analysis and applications, III
1736:Publications Mathématiques de l'Université Paris
447:The starting point of the program may be seen as
1932:, Annals of Mathematics Studies, vol. 151,
510:-functions constitutes Artin's reciprocity law.
2110:Laumon, G.; Rapoport, M.; Stuhler, U. (1993), "
1302:topological construction of algebraic varieties
762:and every finite-dimensional representation of
703:this is expected to give a parameterization of
902:The geometric Langlands program, suggested by
880:) is the field of rational functions over the
292:), the work and approach of Harish-Chandra on
172:. This is accomplished through abstraction to
2274:
2070:"Problems in the theory of automorphic forms"
1852:Bulletin of the American Mathematical Society
1788:Bulletin of the American Mathematical Society
1491:"Proof of the geometric Langlands conjecture"
1390:"Math Quartet Joins Forces on Unified Theory"
1335:ways, providing potential exact solutions in
1292:for the posited objects exists, and if their
746:can be used. Furthermore, given such a group
8:
1732:"Les débuts d'une formule des traces stable"
1619:Journal of the American Mathematical Society
1176:
1161:) for positive characteristic local fields
1065:{\displaystyle {\text{GL}}(2,\mathbb {Q} )}
227:, the Langlands program allows a potential
168:to the automorphic forms under which it is
93:is a web of far-reaching and consequential
2281:
2267:
2259:
1458:Love and Math: The Heart of Hidden Reality
2214:
2176:. Cambridge: Cambridge University Press.
1926:Harris, Michael; Taylor, Richard (2001),
1896:
1863:
1799:
1766:
1698:
1630:
1584:
1541:
1519:
1055:
1054:
1040:
1038:
1002:
1001:
999:
976:
975:
973:
820:
819:
817:
631:
630:
628:
605:
604:
602:
563:
562:
560:
402:, whose existence is unproven, or on the
114:
110:
69:Learn how and when to remove this message
1830:An Introduction to the Langlands Program
1196:
941:A 9-person collaborative project led by
838:(the original and most important case),
32:This article includes a list of general
1381:
1204:
581:). Langlands then generalized these to
289:
1165:. Their proof uses a global argument.
1124:
1024:of their irreducible representations.
1298:rational solutions of elliptic curves
1217:Fundamental lemma (Langlands program)
7:
2463:Birch and Swinnerton-Dyer conjecture
2174:The Genesis of the Langlands Program
1687:Publications Mathématiques de l'IHÉS
1269:allowing an exact quantification of
1187:) for characteristic 0 local fields
957:The Langlands conjectures for GL(1,
583:automorphic cuspidal representations
490:-functions are identical to certain
2549:Representation theory of Lie groups
1131:for the general linear group GL(2,
1742:(13). Paris: Université de Paris.
1265:constructions results in powerful
898:Geometric Langlands correspondence
347:but also had a meaning visible in
280:formulated a few years earlier by
38:it lacks sufficient corresponding
14:
2508:Main conjecture of Iwasawa theory
1560:. Quanta Magazine. July 19, 2024.
254:which underpin numbers and their
223:. As an analogue to the possible
176:, by an equivalence to a certain
2022:"The Langlands Conjecture for Gl
1365:Jacquet–Langlands correspondence
23:
1865:10.1090/S0273-0979-1984-15237-6
1312:, each of which relates to the
552:on the upper half plane of the
188:. Consequently, this allows an
2442:Ramanujan–Petersson conjecture
2432:Generalized Riemann hypothesis
2328:-functions of Hecke characters
1059:
1045:
842:, and function fields (finite
540:had earlier related Dirichlet
343:idea came out of the cusps on
174:higher dimensional integration
121:in algebraic number theory to
1:
2401:Analytic class number formula
2254:The work of Robert Langlands
1801:10.1090/S0273-0979-02-00963-1
1730:Langlands, Robert P. (1983).
1310:generalized analytical series
914:-adic representations of the
641:{\displaystyle \mathbb {Q} ,}
517:-functions in a natural way:
296:, and in technical terms the
2406:Riemann–von Mangoldt formula
1327:have been posited, as their
1277:and further abstractions of
1009:{\displaystyle \mathbb {C} }
983:{\displaystyle \mathbb {R} }
827:{\displaystyle \mathbb {Q} }
785:between their corresponding
612:{\displaystyle \mathbb {Q} }
570:{\displaystyle \mathbb {C} }
225:exact distribution of primes
1734:. U.E.R. de Mathématiques.
1461:, Basic Books, p. 77,
1253:between abstract algebraic
1129:local Langlands conjectures
1116:local Langlands conjectures
1110:Local Langlands conjectures
750:, Langlands constructs the
2582:
2058:Langlands, Robert (1967),
1934:Princeton University Press
1662:Stroh, B. (January 2016).
1339:(as was similarly done in
1214:
1113:
895:
194:invariance transformations
158:fundamental representation
97:about connections between
2233:10.1007/s00222-012-0420-5
2068:Langlands, R. P. (1970),
1709:10.1007/s10240-010-0026-7
807:restricted representation
734:Generalized functoriality
262:which base them. Through
192:construction of powerful
2202:Inventiones Mathematicae
2117:Inventiones Mathematicae
1969:Inventiones Mathematicae
1022:Langlands classification
712:Langlands classification
355:", contrasted with the "
277:philosophy of cusp forms
2358:Dedekind zeta functions
2020:Kutzko, Philip (1980),
1906:Gelfand, I. M. (1963),
1245:and generalizations of
916:Ă©tale fundamental group
836:algebraic number fields
87:algebraic number theory
53:more precise citations.
2564:History of mathematics
1832:. Boston: Birkhäuser.
1613:Lafforgue, V. (2018).
1571:Lafforgue, V. (2018).
1243:analytic number theory
1207:) gave another proof.
1091:) for function fields
1066:
1010:
984:
949:as part of the proof.
828:
795:induced representation
696:-group are not fixed.
676:reciprocity conjecture
642:
613:
571:
469:algebraic number field
239:objects through their
217:fundamental structures
117:), it seeks to relate
2478:Bloch–Kato conjecture
2473:Beilinson conjectures
2456:Algebraic conjectures
2311:Riemann zeta function
2031:Annals of Mathematics
1990:10.1007/s002220050012
1275:diophantine equations
1261:and their analytical
1135:) over local fields.
1067:
1011:
985:
930:-adic sheaves on the
892:Geometric conjectures
829:
643:
614:
577:that satisfy certain
572:
550:holomorphic functions
500:Riemann zeta function
461:Artin reciprocity law
457:quadratic reciprocity
379:Hilbert modular forms
294:semisimple Lie groups
241:automorphic functions
190:analytical functional
127:representation theory
83:representation theory
2544:Zeta and L-functions
2483:Langlands conjecture
2468:Deligne's conjecture
2420:Analytic conjectures
2061:Letter to Prof. Weil
1294:analytical functions
1288:of such generalized
1284:Furthermore, if the
1037:
998:
972:
816:
627:
601:
587:general linear group
579:functional equations
559:
554:complex number plane
455:, which generalizes
383:Siegel modular forms
233:algebraic structures
211:in constructing the
147:grand unified theory
107:Robert Langlands
2437:Lindelöf hypothesis
2225:2013InMat.192..663S
2130:1993InMat.113..217L
1982:2000InMat.139..439H
1398:. December 8, 2015.
1345:monstrous moonshine
1320:of number fields.
1279:algebraic functions
1271:prime distributions
1081:Lafforgue's theorem
906:following ideas of
801:had been called a '
742:), other connected
658:Langlands attached
502:) constructed from
413:Representations of
372:Levi decompositions
367:are more numerous.
365:parabolic subgroups
357:continuous spectrum
202:algebraic structure
2427:Riemann hypothesis
2351:Algebraic examples
2138:10.1007/BF01244308
2086:10.1007/BFb0079065
2080:, pp. 18–61,
1593:"alternate source"
1353:analytical methods
1337:superstring theory
1306:Riemann hypothesis
1247:algebraic geometry
1169:Michael Harris
1072:remains unproved.
1062:
1006:
980:
963:class field theory
922:to objects of the
824:
638:
609:
567:
326:class field theory
219:for virtually any
182:absolute extension
2554:Automorphic forms
2539:Langlands program
2526:
2525:
2304:Analytic examples
2183:978-1-108-71094-7
2170:Shahidi, Freydoon
2095:978-3-540-05284-5
2026:of a Local Field"
1943:978-0-691-09090-0
1839:978-3-7643-3211-2
1512:Gaitsgory, Dennis
1487:Gaitsgory, Dennis
1427:978-0-465-05074-1
1304:, and the famous
1227:fundamental lemma
1211:Fundamental lemma
1201:Peter Scholze
1139:GĂ©rard Laumon
1121:Philip Kutzko
1104:Vincent Lafforgue
1077:Laurent Lafforgue
1043:
908:Vladimir Drinfeld
799:automorphic forms
546:automorphic forms
361:Eisenstein series
353:discrete spectrum
209:analytical method
154:fundamental lemma
149:of mathematics."
123:automorphic forms
91:Langlands program
79:
78:
71:
2571:
2447:Artin conjecture
2411:Weil conjectures
2283:
2276:
2269:
2260:
2243:
2218:
2187:
2168:Mueller, Julia;
2164:
2106:
2064:
2054:
2016:
1954:
1922:
1902:
1900:
1884:
1867:
1848:Gelbart, Stephen
1843:
1820:
1803:
1773:
1772:
1770:
1758:
1752:
1751:
1727:
1721:
1720:
1702:
1679:
1673:
1672:
1670:
1659:
1653:
1652:
1641:10.1090/jams/897
1634:
1610:
1604:
1603:
1597:
1590:
1588:
1568:
1562:
1561:
1554:
1548:
1547:
1545:
1532:
1526:
1525:
1523:
1508:
1502:
1501:
1499:
1497:
1483:
1477:
1476:
1449:
1443:
1442:
1419:
1406:
1400:
1399:
1386:
1370:Erlangen program
1267:functional tools
1193:Guy Henniart
1143:Michael Rapoport
1099:) in the 1980s.
1071:
1069:
1068:
1063:
1058:
1044:
1041:
1020:) by giving the
1015:
1013:
1012:
1007:
1005:
989:
987:
986:
981:
979:
947:Hecke eigensheaf
943:Dennis Gaitsgory
938:over the curve.
929:
924:derived category
913:
833:
831:
830:
825:
823:
766:, he defines an
744:reductive groups
647:
645:
644:
639:
634:
621:rational numbers
618:
616:
615:
610:
608:
576:
574:
573:
568:
566:
544:-functions with
504:Hecke characters
465:Galois extension
415:reductive groups
400:Langlands groups
178:analytical group
131:algebraic groups
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49:this article by
40:inline citations
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2199:-adic fields",
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1962:) sur un corps
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1416:Love & Math
1410:Frenkel, Edward
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1255:representations
1251:'Functoriality'
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1018:complex numbers
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320:(or reductive)
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2209:(3): 663–715,
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2017:
1976:(2): 439–455,
1955:
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1898:hep-th/0512172
1885:
1858:(2): 177–219,
1854:, New Series,
1844:
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1790:, New Series,
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1249:, the idea of
1234:
1231:
1215:Main article:
1212:
1209:
1173:Richard Taylor
1147:Ulrich Stuhler
1145:, and
1114:Main article:
1111:
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978:
954:
953:Current status
951:
936:vector bundles
896:Main article:
893:
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345:modular curves
332:) for general
282:Harish-Chandra
271:
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145:as "a kind of
143:Edward Frenkel
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2518:Euler system
2513:Selmer group
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2334:
2333:Automorphic
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1973:
1967:
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1911:
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1794:(1): 39–53,
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1600:math.cnrs.fr
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1494:. Retrieved
1481:
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1437:
1433:
1431:
1415:
1404:
1393:
1384:
1349:
1341:group theory
1322:
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1240:
1236:
1233:Implications
1225:proved the "
1223:Ngô Bảo Châu
1220:
1188:
1184:
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1162:
1158:
1154:
1137:
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1031:Andrew Wiles
1029:
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992:real numbers
967:
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932:moduli stack
901:
885:
882:finite field
877:
872:
868:
860:
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840:local fields
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701:local fields
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672:number field
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660:automorphic
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473:Galois group
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387:theta-series
376:
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340:
338:
333:
329:
315:
308:
306:
304:and others.
275:
273:
256:abstractions
248:characterise
245:
229:general tool
221:number field
206:
198:number field
162:finite field
151:
135:local fields
90:
80:
65:
56:
37:
2559:Conjectures
2372:Hasse–Weil
1826:Gelbart, S.
1625:: 719–891.
1577:icm2018.org
1331:connect in
1286:reciprocity
888:elements).
593:) over the
443:Reciprocity
435:Conjectures
215:mapping of
200:to its own
95:conjectures
51:introducing
2533:Categories
2500:-functions
2385:-functions
2376:-functions
2367:-functions
2337:-functions
2320:-functions
2316:Dirichlet
2293:-functions
1966:-adique",
1780:References
1768:1509.00797
1586:1803.03791
1543:2405.03648
1521:2405.03599
1496:August 19,
1474:phenomena.
1333:nontrivial
1318:structures
1314:invariance
844:extensions
664:-functions
595:adele ring
534:-functions
530:Dirichlet
523:-functions
492:Dirichlet
484:-functions
449:Emil Artin
318:semisimple
270:Background
264:analytical
260:invariants
252:structures
237:arithmetic
34:references
2216:1010.1540
2162:124557672
2146:0020-9910
2014:120799103
1998:0020-9910
1874:0002-9904
1810:0002-9904
1717:118103635
1700:0801.0446
1693:: 1–169.
1649:118317537
1632:1209.5352
1436:-branes,
1329:dualities
1221:In 2008,
1102:In 2018,
1075:In 1998,
341:cusp form
322:Lie group
266:methods.
213:categoric
170:invariant
164:with its
59:June 2022
2394:Theorems
2381:Motivic
2241:15124490
1455:(2013),
1412:(2013).
1359:See also
1343:through
1325:M theory
1290:algebras
859:) where
783:morphism
708:-packets
336:> 2.
103:geometry
2221:Bibcode
2195:) over
2154:1228127
2126:Bibcode
2104:0302614
2052:1971151
2006:1738446
1978:Bibcode
1952:1876802
1920:0175997
1882:0733692
1818:1943132
1748:0697567
1316:within
1203: (
1195: (
1175: (
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1079:proved
803:lifting
496:-series
477:abelian
393:Objects
359:" from
302:Selberg
288: (
286:Gelfand
186:algebra
184:of its
109: (
47:improve
2496:-adic
2363:Artin
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918:of an
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685:to an
519:Artin
471:whose
467:of an
459:. The
385:, and
196:for a
180:as an
139:adeles
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36:, but
2237:S2CID
2211:arXiv
2158:S2CID
2048:JSTOR
2010:S2CID
1893:arXiv
1763:arXiv
1713:S2CID
1695:arXiv
1669:(PDF)
1645:S2CID
1627:arXiv
1596:(PDF)
1581:arXiv
1538:arXiv
1516:arXiv
1376:Notes
1263:prime
1016:(the
990:(the
884:with
865:prime
863:is a
699:Over
619:(the
538:Hecke
160:of a
133:over
2178:ISBN
2142:ISSN
2090:ISBN
1994:ISSN
1938:ISBN
1870:ISSN
1834:ISBN
1806:ISSN
1498:2024
1463:ISBN
1422:ISBN
1205:2013
1197:2000
1177:2001
1151:1993
1125:1980
867:and
648:see
398:the
351:as "
339:The
290:1963
284:and
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137:and
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855:(
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764:G
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334:n
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