36:
1044:' 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
408:
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
254:. 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.
1361:
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
1484:
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
741:-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.
1248:
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.
1546:
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".
1451:-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.
318:
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
218:
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
1081:
335:, 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
1038:
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.
1117:
established the global
Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields.
657:
1025:
999:
843:
628:
586:
163:
The
Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. To oversimplify, the
921:, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates
1240:", which was originally conjectured by Langlands and Shelstad in 1983 and being required in the proof of some important conjectures in the Langlands program.
692:
Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a
721:
of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers, this correspondence is the
257:
In short, a simplified description for this theory to a non-specialist would be: the construct of a generalised and somewhat unified framework, to
417:-group that has several inequivalent definitions. Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1967.
2559:
2291:
1568:
2488:
450:
There are several different ways of stating the
Langlands conjectures, which are closely related but not obviously equivalent.
2192:
2104:
1952:
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1436:
1237:
1227:
164:
2473:
1375:
1308:
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Automorphic forms on reductive groups over global fields (with subcases corresponding to number fields or function fields).
17:
1307:
can be shown to be well-defined, some very deep results in mathematics could be within reach of proof. Examples include:
593:
152:. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by
2335:
804:-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an
2452:
1400:
908:
792:
He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved)
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79:
57:
50:
2416:
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2457:
2442:
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367:
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In all these approaches there was no shortage of technical methods, often inductive in nature and based on
820:
is contravariant). Attempts to specify a direct construction have only produced some conditional results.
800:-groups, this conjecture relates their automorphic representations in a way that is compatible with their
438:
Finite fields. Langlands did not originally consider this case, but his conjectures have analogues for it.
2277:
1139:
1126:
1525:; Raskin, Sam (May 2024). "Proof of the geometric Langlands conjecture I: construction of the functor".
2564:
2549:
1944:
1343:
1261:
258:
168:
1772:
Milne, James (2015-09-02). "The Riemann Hypothesis over Finite Fields: From Weil to the Present Day".
926:
817:
1443:
All this stuff, as my dad put it, is quite heavy: we've got Hitchin moduli spaces, mirror symmetry,
2212:
2127:
1979:
1032:
722:
637:
524:
For non-abelian Galois groups and higher-dimensional representations of them, one can still define
287:
44:
1008:
982:
826:
611:
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385:
amongst other matters, but the field was – and is – very demanding.
1603:
956:
has announced a proof of the (categorical, unramified) geometric Langlands conjecture leveraging
942:
327:
For example, in the work of Harish-Chandra one finds the principle that what can be done for one
97:
703:. There are numerous variations of this, in part because the definitions of Langlands group and
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2343:
1676:
La paramétrisation de Langlands globale sur les corps des fonctions (d'après Vincent Lafforgue)
1281:
1253:
1091:
846:
805:
670:
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247:
61:
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Langlands generalized the idea of functoriality: instead of using the general linear group GL(
2483:
2321:
2041:
1693:
1467:
1339:
1296:
1285:
1233:
510:
471:
467:
308:
270:
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192:
93:
2569:
2411:
2326:
2231:
2164:
2136:
2114:
2016:
1988:
1962:
1930:
1892:
1828:
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697:
685:
is equal to one arising from an automorphic cuspidal representation. This is known as his "
597:
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8:
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1992:
979:
Langlands proved the Langlands conjectures for groups over the archimedean local fields
420:
There are different types of objects for which the Langlands conjectures can be stated:
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2221:
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2058:
2020:
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1723:
1705:
1655:
1637:
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1548:
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1347:
1316:
1257:
973:
813:
547:, which would allow the formulation of Artin's statement in this more general setting.
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382:
336:
239:
235:
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over local fields (with different subcases corresponding to archimedean local fields,
2503:
2188:
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2152:
2100:
2024:
2004:
1948:
1880:
1844:
1816:
1727:
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1432:
1300:
1273:
1114:
1087:
918:
463:
371:
227:
2251:
1875:
1106:. This work continued earlier investigations by Drinfeld, who proved the case GL(2,
681:-function arising from a finite-dimensional representation of the Galois group of a
2421:
2373:
2239:
2180:
2144:
2092:
2050:
1996:
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1806:
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1647:
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1497:
1380:
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1328:
1324:
1320:
1312:
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957:
953:
934:
809:
785:-functions satisfy a certain functional equation generalizing those of other known
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529:
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to the one-dimensional representations of this Galois group, and states that these
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266:
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914:
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2012:
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686:
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514:
425:
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359:
141:
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1501:
2269:
1899:
1584:"Shtukas for reductive groups and Langlands correspondence for function fields"
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1420:
1157:
1028:
762:
564:
397:
375:
355:
296:
292:
153:
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1969:
Henniart, Guy (2000), "Une preuve simple des conjectures de Langlands pour GL(
1719:
1682:. Séminaire Bourbaki 68ème année, 2015–2016, no. 1110, Janvier 2016.
2543:
2352:
2307:
2156:
2080:
2070:
2008:
1884:
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unification of many distant mathematical fields into a formalism of powerful
1211:
1131:
946:
660:
487:
109:
1319:. Such proofs would be expected to utilize abstract solutions in objects of
823:
All these conjectures can be formulated for more general fields in place of
596:, which are certain infinite dimensional irreducible representations of the
2528:
2523:
1351:
1269:
1265:
1203:
1041:
892:
875:
726:
682:
483:
312:
231:
212:
208:
176:
172:
129:
2000:
1190:) proved the local Langlands conjectures for the general linear group GL(
1164:) proved the local Langlands conjectures for the general linear group GL(
1002:
850:
711:
548:
145:
1902:(2005). "Lectures on the Langlands Program and Conformal Field Theory".
2300:
2148:
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2062:
1908:
1651:
1304:
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to these automorphic representations, and conjectured that every Artin
605:
491:
459:
200:
149:
105:
1918:
441:
More general fields, such as function fields over the complex numbers.
1626:"Chtoucas pour les groupes réductifs et paramétrisation de Langlands"
332:
2054:
1094:
verifying the Langlands conjectures for the general linear group GL(
737:
The functoriality conjecture states that a suitable homomorphism of
374:. It becomes much more technical for bigger Lie groups, because the
2264:
1778:
1596:
1553:
1531:
1335:
793:
715:
634:). (This ring simultaneously keeps track of all the completions of
113:
2226:
2202:
Scholze, Peter (2013), "The Local Langlands Correspondence for GL(
1710:
1642:
539:
The insight of Langlands was to find the proper generalization of
285:
In a very broad context, the program built on existing ideas: the
167:
of the project posits a direct connection between the generalized
1334:
Additionally, some connections between the Langlands program and
320:
1861:(1984), "An elementary introduction to the Langlands program",
517:. The precise correspondence between these different kinds of
388:
And on the side of modular forms, there were examples such as
27:
Far-reaching conjectures connecting number theory and geometry
769:, and then, for every automorphic cuspidal representation of
242:
for the resolution of invariance at the level of generalized
1940:
The geometry and cohomology of some simple Shimura varieties
1284:. This, in turn, yields the capacity for classification of
2087:, Lecture Notes in Math, vol. 170, Berlin, New York:
1545:
816:', known in special cases, and so is covariant (whereas a
729:, it should give a parameterization of automorphic forms.
509:
or more general series (that is, certain analogues of the
1919:"Automorphic functions and the theory of representations"
1696:(2010). "Le lemme fondamental pour les algèbres de Lie".
1569:"Monumental Proof Settles Geometric Langlands Conjecture"
1210:) gave another proof. Both proofs use a global argument.
1797:
Arthur, James (2003), "The principle of functoriality",
1923:
Proc. Internat. Congr. Mathematicians (Stockholm, 1962)
432:-adic local fields, and completions of function fields)
246:. This in turn permits a somewhat unified analysis of
2125:-elliptic sheaves and the Langlands correspondence",
2120:
1161:
1050:
1011:
985:
829:
640:
614:
572:
1252:
As the program posits a powerful connection between
808:
construction—what in the more traditional theory of
781:-function. One of his conjectures states that these
339:, the way was open at least to speculation about GL(
1925:, Djursholm: Inst. Mittag-Leffler, pp. 74–85,
1424:
1075:
1019:
993:
972:) follow from (and are essentially equivalent to)
837:
725:of representations of real reductive groups. Over
651:
622:
580:
2085:Lectures in modern analysis and applications, III
1747:Publications Mathématiques de l'Université Paris
458:The starting point of the program may be seen as
2541:
2299:
1943:, Annals of Mathematics Studies, vol. 151,
1834:
521:-functions constitutes Artin's reciprocity law.
2121:Laumon, G.; Rapoport, M.; Stuhler, U. (1993), "
1521:
1313:topological construction of algebraic varieties
773:and every finite-dimensional representation of
714:this is expected to give a parameterization of
2178:
1120:
913:The geometric Langlands program, suggested by
891:) is the field of rational functions over the
303:), the work and approach of Harish-Chandra on
183:. This is accomplished through abstraction to
2285:
2081:"Problems in the theory of automorphic forms"
1863:Bulletin of the American Mathematical Society
1799:Bulletin of the American Mathematical Society
1502:"Proof of the geometric Langlands conjecture"
1401:"Math Quartet Joins Forces on Unified Theory"
1346:ways, providing potential exact solutions in
1303:for the posited objects exists, and if their
757:can be used. Furthermore, given such a group
1936:
1743:"Les débuts d'une formule des traces stable"
1630:Journal of the American Mathematical Society
1187:
744:
1172:) for positive characteristic local fields
1076:{\displaystyle {\text{GL}}(2,\mathbb {Q} )}
238:, the Langlands program allows a potential
179:to the automorphic forms under which it is
104:is a web of far-reaching and consequential
2292:
2278:
1469:Love and Math: The Heart of Hidden Reality
2225:
2187:. Cambridge: Cambridge University Press.
2078:
2068:
1937:Harris, Michael; Taylor, Richard (2001),
1907:
1874:
1810:
1777:
1740:
1709:
1641:
1623:
1595:
1581:
1552:
1530:
1066:
1013:
987:
831:
642:
616:
574:
413:, whose existence is unproven, or on the
125:
121:
80:Learn how and when to remove this message
1968:
1841:An Introduction to the Langlands Program
1207:
952:A 9-person collaborative project led by
902:
849:(the original and most important case),
43:This article includes a list of general
2201:
1916:
1898:
1857:
1462:
1419:
1215:
592:). Langlands then generalized these to
300:
14:
2542:
2030:
1796:
1176:. Their proof uses a global argument.
1135:
1035:of their irreducible representations.
2273:
1771:
1672:
1496:
1309:rational solutions of elliptic curves
1228:Fundamental lemma (Langlands program)
2474:Birch and Swinnerton-Dyer conjecture
2185:The Genesis of the Langlands Program
1698:Publications Mathématiques de l'IHÉS
1692:
1280:allowing an exact quantification of
1221:
1198:) for characteristic 0 local fields
968:The Langlands conjectures for GL(1,
594:automorphic cuspidal representations
501:-functions are identical to certain
29:
2560:Representation theory of Lie groups
1142:for the general linear group GL(2,
24:
1753:(13). Paris: Université de Paris.
1276:constructions results in powerful
909:Geometric Langlands correspondence
358:but also had a meaning visible in
291:formulated a few years earlier by
49:it lacks sufficient corresponding
25:
2586:
2519:Main conjecture of Iwasawa theory
2258:
1571:. Quanta Magazine. July 19, 2024.
963:
265:which underpin numbers and their
234:. As an analogue to the possible
187:, by an equivalence to a certain
2033:"The Langlands Conjecture for Gl
1376:Jacquet–Langlands correspondence
732:
34:
1876:10.1090/S0273-0979-1984-15237-6
1765:
1734:
1686:
1666:
1323:, each of which relates to the
1243:
563:on the upper half plane of the
199:. Consequently, this allows an
2453:Ramanujan–Petersson conjecture
2443:Generalized Riemann hypothesis
2339:-functions of Hecke characters
1617:
1575:
1561:
1539:
1515:
1490:
1456:
1413:
1393:
1070:
1056:
853:, and function fields (finite
551:had earlier related Dirichlet
453:
445:
354:idea came out of the cusps on
185:higher dimensional integration
132:in algebraic number theory to
13:
1:
2412:Analytic class number formula
2265:The work of Robert Langlands
1812:10.1090/S0273-0979-02-00963-1
1790:
1741:Langlands, Robert P. (1983).
1321:generalized analytical series
925:-adic representations of the
652:{\displaystyle \mathbb {Q} ,}
528:-functions in a natural way:
307:, and in technical terms the
280:
2417:Riemann–von Mangoldt formula
1338:have been posited, as their
1288:and further abstractions of
1020:{\displaystyle \mathbb {C} }
994:{\displaystyle \mathbb {R} }
838:{\displaystyle \mathbb {Q} }
796:between their corresponding
623:{\displaystyle \mathbb {Q} }
581:{\displaystyle \mathbb {C} }
236:exact distribution of primes
7:
1745:. U.E.R. de Mathématiques.
1472:, Basic Books, p. 77,
1369:
1264:between abstract algebraic
1140:local Langlands conjectures
1127:local Langlands conjectures
1121:Local Langlands conjectures
761:, Langlands constructs the
10:
2591:
2069:Langlands, Robert (1967),
1945:Princeton University Press
1673:Stroh, B. (January 2016).
1350:(as was similarly done in
1225:
1124:
906:
403:
205:invariance transformations
169:fundamental representation
108:about connections between
2502:
2466:
2430:
2404:
2361:
2314:
2244:10.1007/s00222-012-0420-5
2079:Langlands, R. P. (1970),
1720:10.1007/s10240-010-0026-7
818:restricted representation
745:Generalized functoriality
273:which base them. Through
203:construction of powerful
2213:Inventiones Mathematicae
2128:Inventiones Mathematicae
1980:Inventiones Mathematicae
1386:
1033:Langlands classification
723:Langlands classification
366:", contrasted with the "
288:philosophy of cusp forms
2369:Dedekind zeta functions
2031:Kutzko, Philip (1980),
1917:Gelfand, I. M. (1963),
1256:and generalizations of
927:Ă©tale fundamental group
847:algebraic number fields
98:algebraic number theory
64:more precise citations.
2575:History of mathematics
1843:. Boston: Birkhäuser.
1624:Lafforgue, V. (2018).
1582:Lafforgue, V. (2018).
1254:analytic number theory
1218:) gave another proof.
1102:) for function fields
1077:
1021:
995:
960:as part of the proof.
839:
806:induced representation
707:-group are not fixed.
687:reciprocity conjecture
653:
624:
582:
480:algebraic number field
250:objects through their
228:fundamental structures
128:), it seeks to relate
2489:Bloch–Kato conjecture
2484:Beilinson conjectures
2467:Algebraic conjectures
2322:Riemann zeta function
2042:Annals of Mathematics
2001:10.1007/s002220050012
1286:diophantine equations
1272:and their analytical
1146:) over local fields.
1078:
1022:
996:
941:-adic sheaves on the
903:Geometric conjectures
840:
654:
625:
588:that satisfy certain
583:
561:holomorphic functions
511:Riemann zeta function
472:Artin reciprocity law
468:quadratic reciprocity
390:Hilbert modular forms
305:semisimple Lie groups
252:automorphic functions
201:analytical functional
138:representation theory
94:representation theory
2555:Zeta and L-functions
2494:Langlands conjecture
2479:Deligne's conjecture
2431:Analytic conjectures
2072:Letter to Prof. Weil
1305:analytical functions
1299:of such generalized
1295:Furthermore, if the
1048:
1009:
983:
827:
638:
612:
598:general linear group
590:functional equations
570:
565:complex number plane
466:, which generalizes
394:Siegel modular forms
244:algebraic structures
222:in constructing the
158:grand unified theory
118:Robert Langlands
18:Langlands philosophy
2448:Lindelöf hypothesis
2236:2013InMat.192..663S
2141:1993InMat.113..217L
1993:2000InMat.139..439H
1409:. December 8, 2015.
1356:monstrous moonshine
1331:of number fields.
1290:algebraic functions
1282:prime distributions
1092:Lafforgue's theorem
917:following ideas of
812:had been called a '
753:), other connected
669:Langlands attached
513:) constructed from
424:Representations of
383:Levi decompositions
378:are more numerous.
376:parabolic subgroups
368:continuous spectrum
213:algebraic structure
2438:Riemann hypothesis
2362:Algebraic examples
2149:10.1007/BF01244308
2097:10.1007/BFb0079065
2091:, pp. 18–61,
1604:"alternate source"
1364:analytical methods
1348:superstring theory
1317:Riemann hypothesis
1258:algebraic geometry
1180:Michael Harris
1083:remains unproved.
1073:
1017:
991:
974:class field theory
933:to objects of the
835:
649:
620:
578:
337:class field theory
230:for virtually any
193:absolute extension
2565:Automorphic forms
2550:Langlands program
2537:
2536:
2315:Analytic examples
2194:978-1-108-71094-7
2181:Shahidi, Freydoon
2106:978-3-540-05284-5
2037:of a Local Field"
1954:978-0-691-09090-0
1850:978-3-7643-3211-2
1523:Gaitsgory, Dennis
1498:Gaitsgory, Dennis
1438:978-0-465-05074-1
1315:, and the famous
1238:fundamental lemma
1222:Fundamental lemma
1212:Peter Scholze
1150:GĂ©rard Laumon
1132:Philip Kutzko
1115:Vincent Lafforgue
1088:Laurent Lafforgue
1054:
919:Vladimir Drinfeld
810:automorphic forms
557:automorphic forms
372:Eisenstein series
364:discrete spectrum
220:analytical method
165:fundamental lemma
160:of mathematics."
134:automorphic forms
102:Langlands program
90:
89:
82:
16:(Redirected from
2582:
2458:Artin conjecture
2422:Weil conjectures
2294:
2287:
2280:
2271:
2270:
2254:
2229:
2198:
2179:Mueller, Julia;
2175:
2117:
2075:
2065:
2027:
1965:
1933:
1913:
1911:
1895:
1878:
1859:Gelbart, Stephen
1854:
1831:
1814:
1784:
1783:
1781:
1769:
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1762:
1738:
1732:
1731:
1713:
1690:
1684:
1683:
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1670:
1664:
1663:
1652:10.1090/jams/897
1645:
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1601:
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1508:
1494:
1488:
1487:
1460:
1454:
1453:
1430:
1417:
1411:
1410:
1397:
1381:Erlangen program
1278:functional tools
1204:Guy Henniart
1154:Michael Rapoport
1110:) in the 1980s.
1082:
1080:
1079:
1074:
1069:
1055:
1052:
1031:) by giving the
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958:Hecke eigensheaf
954:Dennis Gaitsgory
949:over the curve.
940:
935:derived category
924:
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842:
841:
836:
834:
777:, he defines an
755:reductive groups
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587:
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555:-functions with
515:Hecke characters
476:Galois extension
426:reductive groups
411:Langlands groups
189:analytical group
142:algebraic groups
85:
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71:
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60:this article by
51:inline citations
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2210:-adic fields",
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2107:
2089:Springer-Verlag
2055:10.2307/1971151
2036:
1973:) sur un corps
1955:
1900:Frenkel, Edward
1851:
1839:, eds. (2003).
1835:Bernstein, J.;
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1427:Love & Math
1421:Frenkel, Edward
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1266:representations
1262:'Functoriality'
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331:(or reductive)
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177:group extension
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56:Please help to
55:
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2220:(3): 663–715,
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1987:(2): 439–455,
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1869:(2): 177–219,
1865:, New Series,
1855:
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1260:, the idea of
1245:
1242:
1226:Main article:
1223:
1220:
1184:Richard Taylor
1158:Ulrich Stuhler
1156:, and
1125:Main article:
1122:
1119:
1072:
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1061:
1058:
1015:
989:
965:
964:Current status
962:
947:vector bundles
907:Main article:
904:
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763:Langlands dual
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356:modular curves
343:) for general
293:Harish-Chandra
282:
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156:as "a kind of
154:Edward Frenkel
116:. Proposed by
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1138:) proved the
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915:GĂ©rard Laumon
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664:-adic numbers
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309:trace formula
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2529:Euler system
2524:Selmer group
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2344:Automorphic
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1978:
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1805:(1): 39–53,
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1611:math.cnrs.fr
1610:
1587:
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1541:
1517:
1505:. Retrieved
1492:
1483:
1468:
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1444:
1442:
1426:
1415:
1404:
1395:
1360:
1352:group theory
1333:
1294:
1251:
1247:
1244:Implications
1236:proved the "
1234:Ngô Bảo Châu
1231:
1199:
1195:
1191:
1178:
1173:
1169:
1165:
1148:
1143:
1130:
1112:
1107:
1103:
1099:
1095:
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1042:Andrew Wiles
1040:
1037:
1003:real numbers
978:
969:
967:
951:
943:moduli stack
912:
896:
893:finite field
888:
883:
879:
871:
867:
862:
858:
851:local fields
822:
801:
797:
791:
789:-functions.
786:
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748:
738:
736:
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712:local fields
709:
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683:number field
678:
672:
671:automorphic
668:
661:
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484:Galois group
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419:
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398:theta-series
387:
380:
351:
349:
344:
340:
326:
319:
317:
315:and others.
286:
284:
267:abstractions
259:characterise
256:
240:general tool
232:number field
217:
209:number field
173:finite field
162:
146:local fields
101:
91:
76:
67:
48:
2570:Conjectures
2383:Hasse–Weil
1837:Gelbart, S.
1636:: 719–891.
1588:icm2018.org
1342:connect in
1297:reciprocity
899:elements).
604:) over the
454:Reciprocity
446:Conjectures
226:mapping of
211:to its own
106:conjectures
62:introducing
2544:Categories
2511:-functions
2396:-functions
2387:-functions
2378:-functions
2348:-functions
2331:-functions
2327:Dirichlet
2304:-functions
1977:-adique",
1791:References
1779:1509.00797
1597:1803.03791
1554:2405.03648
1532:2405.03599
1507:August 19,
1485:phenomena.
1344:nontrivial
1329:structures
1325:invariance
855:extensions
675:-functions
606:adele ring
545:-functions
541:Dirichlet
534:-functions
503:Dirichlet
495:-functions
460:Emil Artin
329:semisimple
281:Background
275:analytical
271:invariants
263:structures
248:arithmetic
45:references
2227:1010.1540
2173:124557672
2157:0020-9910
2025:120799103
2009:0020-9910
1885:0002-9904
1821:0002-9904
1728:118103635
1711:0801.0446
1704:: 1–169.
1660:118317537
1643:1209.5352
1447:-branes,
1340:dualities
1232:In 2008,
1113:In 2018,
1086:In 1998,
352:cusp form
333:Lie group
277:methods.
224:categoric
181:invariant
175:with its
70:June 2022
2405:Theorems
2392:Motivic
2252:15124490
1466:(2013),
1423:(2013).
1370:See also
1354:through
1336:M theory
1301:algebras
870:) where
794:morphism
719:-packets
347:> 2.
114:geometry
2232:Bibcode
2206:) over
2165:1228127
2137:Bibcode
2115:0302614
2063:1971151
2017:1738446
1989:Bibcode
1963:1876802
1931:0175997
1893:0733692
1829:1943132
1759:0697567
1327:within
1214: (
1206: (
1186: (
1160: (
1134: (
1090:proved
814:lifting
507:-series
488:abelian
404:Objects
370:" from
313:Selberg
299: (
297:Gelfand
197:algebra
195:of its
120: (
58:improve
2507:-adic
2374:Artin
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2171:
2163:
2155:
2113:
2103:
2061:
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1406:Quanta
1152:,
1005:) and
929:of an
765:group
701:-group
696:to an
530:Artin
482:whose
478:of an
470:. The
396:, and
207:for a
191:as an
150:adeles
100:, the
47:, but
2248:S2CID
2222:arXiv
2169:S2CID
2059:JSTOR
2021:S2CID
1904:arXiv
1774:arXiv
1724:S2CID
1706:arXiv
1680:(PDF)
1656:S2CID
1638:arXiv
1607:(PDF)
1592:arXiv
1549:arXiv
1527:arXiv
1387:Notes
1274:prime
1027:(the
1001:(the
895:with
876:prime
874:is a
710:Over
630:(the
549:Hecke
171:of a
144:over
2189:ISBN
2153:ISSN
2101:ISBN
2005:ISSN
1949:ISBN
1881:ISSN
1845:ISBN
1817:ISSN
1509:2024
1474:ISBN
1433:ISBN
1216:2013
1208:2000
1188:2001
1162:1993
1136:1980
878:and
659:see
409:the
362:as "
350:The
301:1963
295:and
261:the
148:and
136:and
126:1970
122:1967
112:and
96:and
2306:in
2240:doi
2218:192
2145:doi
2133:113
2093:doi
2051:doi
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1997:doi
1985:139
1871:doi
1807:doi
1751:VII
1716:doi
1702:111
1648:doi
1358:).
1268:of
945:of
937:of
857:of
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608:of
600:GL(
486:is
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311:of
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2013:MR
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1959:MR
1957:,
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1887:,
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1192:n
1174:K
1170:K
1166:n
1144:K
1108:K
1104:K
1100:K
1096:n
1071:)
1067:Q
1063:,
1060:2
1057:(
1014:C
988:R
970:K
939:l
923:l
897:p
889:t
887:(
884:p
880:F
872:p
868:t
866:(
863:p
859:F
832:Q
802:L
798:L
787:L
783:L
779:L
775:G
771:G
767:G
759:G
751:n
739:L
717:L
705:L
699:L
679:L
673:L
662:p
647:,
643:Q
617:Q
602:n
575:C
559:(
553:L
543:L
532:L
526:L
519:L
505:L
499:L
493:L
430:p
415:L
345:n
341:n
83:)
77:(
72:)
68:(
54:.
20:)
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