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Arithmetic applications of the Langlands program

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Japanese Journal of Mathematics Aims and scope

Abstract.

This expository article is an introduction to the Langlands functoriality conjectures and their applications to the arithmetic of representations of Galois groups of number fields. Thanks to the work of a great many people, the stable trace formula is now largely established in a version adequate for proving Langlands functoriality in the setting of endoscopy. These developments are discussed in the first two sections of the article. The final section describes the compatible families of ℓ-adic Galois representations that can be attached to automorphic forms with the help of Shimura varieties. To illustrate the relevance of Langlands functoriality to number theory, the article concludes with a description of the Sato–Tate conjecture for elliptic modular forms, recently proved in joint work of Barnet-Lamb, Geraghty, Taylor, and the author.

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References

  1. J. Arthur, The invariant trace formula. II. Global theory, J. Amer. Math. Soc., 1 (1988), 501–554.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Arthur, The L 2-Lefschetz numbers of Hecke operators, Invent. Math., 97 (1989), 257–290.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Arthur, A stable trace formula. I. General expansions, J. Inst. Math. Jussieu, 1 (2002), 175–277.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Arthur, An introduction to the trace formula, In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, (eds. J. Arthur, D. Ellwood and R. Kottwitz), Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005, pp. 1–263.

  5. J. Arthur, Lectures at Banff International Research Station, August 2008, videos accessible at http://www.birs.ca/publications/workshop_videos/2008.htm.

  6. J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, book to appear.

  7. J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Stud., 120, Princeton Univ. Press, 1989.

  8. T. Barnet-Lamb, Potential automorphy for certain Galois representations to GL(2n), manuscript, 2008.

  9. T. Barnet-Lamb, On the potential automorphy of certain odd-dimensional Galois representations, manuscript, 2009.

  10. T. Barnet-Lamb, T. Gee and D. Geraghty, The Sato–Tate conjecture for Hilbert modular forms, manuscript, 2009.

  11. T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor, A family of Calabi–Yau varieties and potential automorphy II, manuscript, 2009.

  12. J. Bellaïche and G. Chenevier, Families of Galois Representations and Selmer Groups, Astérisque, 324, Soc. Math. France, 2009.

  13. D. Blasius and J.D. Rogawski, Tate classes and arithmetic quotients of the two-ball, in [LR], 421–444.

  14. D. Blasius and J.D. Rogawski, Motives for Hilbert modular forms, Invent. Math., 114 (1993), 55–87.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, second ed., Math. Surveys Monogr., 67, Amer. Math. Soc., Providence, RI, 1999.

  16. C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over \({\mathbb Q}\) : wild 3-adic exercises, J. Amer. Math. Soc., 14 (2001), 843–939.

    Google Scholar 

  17. F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., 41 (1972), 5–251; II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math., 60 (1984), 197–376.

  18. D. Bump, Spectral theory and the trace formula, In: An Introduction to the Langlands Program, (eds. J. Bernstein and S. Gelbart), Birkhäuser Boston, Boston, MA, 2003, pp. 153–196.

  19. K. Buzzard and T. Gee, The conjectural connections between automorphic representations and Galois representations, preliminary version, fall 2009.

  20. P. Candelas, X.C. de la Ossa, P.S. Green and L. Parkes, An exactly soluble superconformal theory from a mirror pair of Calabi–Yau manifolds, Nuclear Phys. B, 359 (1991), 21–74.

    Article  MATH  MathSciNet  Google Scholar 

  21. H. Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, In: p-adic Monodromy and the Birch and Swinnerton-Dyer conjecture, Contemp. Math., 165, Amer. Math. Soc., Providence, RI, 1994, pp. 213–237.

  22. L. Clozel, M. Harris, J.-P. Labesse and B.C. Ngô, eds., The Stable Trace Formula, Shimura Varieties, and Arithmetic Applications, Book 1, in preparation, preliminary versions available at http://fa.institut.math.jussieu.fr/node/4.

  23. P.-H. Chaudouard, Sur le transfert lisse des intégrales orbitales d’après Waldspurger, chapter in [Book1].

  24. P.-H. Chaudouard and G. Laumon, Le lemme fondamental pondéré. I. Constructions géométriques; II. Énoncés cohomologiques, http://www.math.u-psud.fr/~chaudou.

  25. G. Chenevier, Une application des variétés de Hecke des groupes unitaires, in [Book 2].

  26. G. Chenevier and M. Harris, Construction of automorphic Galois representations. II, chapter in [Book2].

  27. L. Clozel, The fundamental lemma for stable base change, Duke Math. J., 61 (1990), 255–302.

    Google Scholar 

  28. L. Clozel, Représentations Galoisiennes associées aux représentations automorphes autoduales de GL(n), Inst. Hautes Études Sci. Publ. Math., 73 (1991), 97–145.

  29. L. Clozel, M. Harris and J.-P. Labesse, Endoscopic transfer, chapter in [Book1].

  30. L. Clozel, M. Harris and J.-P. Labesse, Construction of automorphic Galois representations. I, chapter in [Book1].

  31. L. Clozel, M. Harris and R. Taylor, Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations, Publ. Math. Inst. Hautes Études Sci., 108 (2008), 1–181.

  32. L. Clozel and J.-P. Labesse, Orbital integrals and distributions, to appear in Shahidi volume.

  33. R. Cluckers, T. Hales and F. Loeser, Transfer principle for the fundamental lemma, chapter in [Book1].

  34. G. Cornell, J. Silverman and G. Stevens, Modular Forms and Fermat’s Last Theorem, Springer-Verlag, 1998.

  35. H. Darmon, F. Diamond and R. Taylor, Fermat’s last theorem, In: Current Developments in Mathematics, 1995, Int. Press, Boston, 1994, pp. 1–154.

  36. J.-F. Dat and N.D. Tuan, Lemme Fondamental pour les algèbres de Lie d’après Ngô Bao–Châu, chapter in [Book1].

  37. P. Deligne, Formes Modulaires et Représentations l-adiques, Séminaire Bourbaki, 11 (1968-1969), Exp. 355.

  38. P. Deligne, Travaux de Shimura, Séminaire Bourbaki, Exp. 389, Lecture Notes in Math., 244, Springer-Verlag, 1971.

  39. F. Diamond, The Taylor–Wiles construction and multiplicity one, Invent. Math., 128 (1997), 379–391.

    Article  MATH  MathSciNet  Google Scholar 

  40. L.V. Dieulefait, Langlands base change for GL(2), manuscript, 2009.

  41. L. Fargues and M. Harris, eds., The Stable Trace Formula, Shimura Varieties, and Arithmetic Applications, Book 2, in preparation.

  42. L. Fargues and E. Mantovan, Variétés de Shimura, Espaces de Rapoport–Zink et Correspondances de Langlands Locales, Astérisque, 291, Soc. Math. France, 2004.

  43. J.-M. Fontaine and B. Mazur, Geometric Galois representations, In: Elliptic Curves, Modular Forms, and Fermat’s Last Theorem, Hong Kong, 1993, Int. Press, Cambridge, MA, 1995, pp. 41–78.

  44. K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, version 2.0, preprint, 1999.

  45. T. Gee, The Sato–Tate conjecture for modular forms of weight 3, manuscript, 2009.

  46. S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Sup. (4), 11 (1978), 471–542.

    Google Scholar 

  47. D. Geraghty, Modularity lifting theorems for ordinary Galois representations, preprint, 2009.

  48. R. Guralnick, M. Harris and N.M. Katz, Automorphic realization of Galois representations, manuscript, 2008.

  49. M. Goresky, R. Kottwitz and R. MacPherson, Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J., 89 (1997), 477–554.

    Article  MATH  MathSciNet  Google Scholar 

  50. L. Guerberoff, Modularity lifting theorems for Galois representations of unitary type, thesis, Univ. Paris 7, in preparation.

  51. T.J. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, (eds. J. Arthur, D. Ellwood and R. Kottwitz), Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005, pp. 583–642.

  52. T.C. Hales, A simple definition of transfer factors for unramified groups, In: Representation Theory of Groups and Algebras, Contemp. Math., 145, Amer. Math. Soc., Providence, RI, 1993, pp. 109–134.

  53. T.C. Hales, On the fundamental lemma for standard endoscopy: reduction to unit elements, Canad. J. Math., 47 (1995), 974–994.

    MATH  MathSciNet  Google Scholar 

  54. M. Harris, Potential automorphy of odd-dimensional symmetric powers of elliptic curves, and applications, In: Algebra, Arithmetic, and Geometry: Volume II: In Honor of Y.I. Manin, (eds. Y. Tschinkel and Yu. Zarhin), Birkhäuser Boston, Boston, MA, to appear.

  55. M. Harris, An introduction to the stable trace formula, chapter in [Book1].

  56. M. Harris, Galois representations, automorphic forms, and the Sato–Tate Conjecture, In: Proceedings of the Clay Research Conferences, 2007-8, to appear.

  57. M. Harris and J.-P. Labesse, Conditional base change for unitary groups, Asian J. Math., 8 (2004), 653–684.

    MATH  MathSciNet  Google Scholar 

  58. M. Harris, N. Shepherd-Barron and R. Taylor, A family of Calabi–Yau varieties and potential automorphy, Ann. of Math. (2), in press.

  59. M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Stud., 151, Princeton Univ. Press, 2001.

  60. G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math., 139 (2000), 439–455.

    Google Scholar 

  61. Y. Ihara, Hecke polynomials as congruence ζ-functions in elliptic modular case, Ann. of Math. (2), 85 (1967), 267–295.

    Article  MathSciNet  Google Scholar 

  62. H. Jacquet and J.A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math., 103 (1981), 499–558; II, Amer. J. Math., 103 (1981), 777–815.

  63. H. Jacquet, I.I. Piatetskii-Shapiro and J.A. Shalika, Rankin–Selberg convolutions, Amer. J. Math., 105 (1983), 367–464.

    Google Scholar 

  64. N.M. Katz, Rigid Local Systems, Ann. of Math. Stud., 139, Princeton Univ. Press, 1996.

  65. N.M. Katz, Another look at the Dwork family, In: Algebra, Arithmetic, and Geometry: Volume II: In Honor of Y.I. Manin, (eds. Y. Tschinkel and Yu. Zarhin), Birkhäuser Boston, Boston, MA, to appear.

  66. C. Khare, M. Larsen and R. Ramakrishna, Transcendental ℓ-adic Galois representations, Math. Res. Lett., 12 (2005), 685–700.

    MATH  MathSciNet  Google Scholar 

  67. H.H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2, J. Amer. Math. Soc., 16 (2003), 139--183.

    Article  MATH  MathSciNet  Google Scholar 

  68. H.H. Kim and F. Shahidi, Functorial products for GL2 ×  GL3 and the symmetric cube for GL2, Ann. of Math. (2), 155 (2002), 837–893.

    Google Scholar 

  69. M. Kisin, Moduli of finite flat group schemes and modularity, Ann. of Math. (2), 170 (2009), 1085–1180.

    Article  MATH  MathSciNet  Google Scholar 

  70. M. Kisin, The Fontaine–Mazur conjecture for GL 2, J. Amer. Math. Soc., 22 (2009), 641–690.

    Article  MathSciNet  Google Scholar 

  71. M. Kisin, Modularity of 2-adic Barsotti–Tate representations, Invent. Math., 178 (2009), 587–634.

    Article  MATH  MathSciNet  Google Scholar 

  72. R.E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J., 51 (1984), 611–650.

    Article  MATH  MathSciNet  Google Scholar 

  73. R.E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann., 275 (1986), 365–399.

    Article  MATH  MathSciNet  Google Scholar 

  74. R.E. Kottwitz, Base change for unit elements of Hecke algebras, Compositio Math., 60 (1986), 237–250.

    MathSciNet  Google Scholar 

  75. R.E. Kottwitz, Tamagawa numbers, Ann. of Math. (2), 127 (1988), 629–646.

    Article  MathSciNet  Google Scholar 

  76. R.E. Kottwitz, Shimura varieties and λ-adic representations, In: Automorphic Forms, Shimura Varieties, and L-functions, Vol. 1, Academic Press, New York, 1990, pp. 161–210.

  77. R.E. Kottwitz, On the λ-adic representations associated to some simple Shimura varieties, Invent. Math., 108 (1992), 653–665.

    Article  MATH  MathSciNet  Google Scholar 

  78. R.E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., 5 (1992), 373–444.

    Article  MATH  MathSciNet  Google Scholar 

  79. R.E. Kottwitz, Transfer factors for Lie algebras., Represent. Theory, 3 (1999), 127–138.

  80. R.E. Kottwitz, Harmonic analysis on reductive p-adic groups and Lie algebras, In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, (eds. J. Arthur, D. Ellwood and R. Kottwitz), Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005, pp. 393–522.

  81. R.E. Kottwitz and D. Shelstad, Foundations of Twisted Endoscopy, Astérisque, 255, Soc. Math. France, 1999.

  82. J.-P. Labesse, Fonctions élémentaires et lemme fondamental pour le changement de base stable, Duke Math. J., 61 (1990), 519–530.

    Article  MATH  MathSciNet  Google Scholar 

  83. J.-P. Labesse, Cohomologie, stabilisation et changement de base, Astérisque, 257 (1999), 1–116.

  84. J.-P. Labesse, Nombres de Tamagawa des groupes réductifs quasi-connexes, Manuscripta Math., 104 (2001), 407–430.

    Article  MATH  MathSciNet  Google Scholar 

  85. J.-P. Labesse, Stable twisted trace formula: elliptic terms, J. Inst. Math. Jussieu, 3 (2004), 473–530.

    Google Scholar 

  86. J.-P. Labesse, Changement de base CM et séries discrètes, chapter in [Book1], with appendix by L. Clozel.

  87. J.-P. Labesse and R.P. Langlands, L-indistinguishability for SL(2), Canad. J. Math., 31 (1979), 726–785.

    Google Scholar 

  88. L. Lafforgue, Chtoucas de Drinfeld, formule des traces d’Arthur–Selberg et correspondance de Langlands, In: Proceedings of the International Congress of Mathematicians, Vol. I, Beijing, 2002, pp. 383–400.

  89. R.P. Langlands, Problems in the theory of automorphic forms, In: Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., 170, Springer-Verlag, 1970, pp. 18–61.

  90. R.P. Langlands, Some contemporary problems with origins in the Jugendtraum, In: Proc. Symp. Pure Math., 28, Amer. Math. Soc., 1976, pp. 401–418.

  91. R.P. Langlands, Automorphic representations, Shimura varieties, and motives, Ein Märchen, In: Proc. Symp. Pure Math., 33, Part 2, Amer. Math. Soc., 1979, pp. 205–246.

  92. R.P. Langlands, Base Change for GL(2), Ann. of Math. Stud., 96, Princeton Univ. Press, 1980.

  93. R.P. Langlands, Les débuts d’une Formule des Traces Stables, Publ. Math. Univ. Paris VII, 13, Univ. Paris VII, 1983.

  94. R.P. Langlands, Beyond endoscopy, In: Contributions to Automorphic Forms, Geometry, and Number Theory: A Volume in Honor of J. Shalika, (eds. H. Hida, D. Ramakrishnan and F. Shahidi), Johns Hopkins Univ. Press, 2004, pp. 611–698.

  95. R.P. Langlands, Un nouveau point de repère dans la théorie des formes automorphes, http://publications.ias.edu/rpl.

  96. R.P. Langlands and D. Ramakrishnan, eds., The Zeta Functions of Picard Modular Surfaces, Univ. de Montréal, Centre de Recherches Mathé matiques, Montreal, 1992.

  97. R.P. Langlands and M. Rapoport, Shimuravarietäten und Gerben, J. Reine Angew. Math., 378 (1987), 113–220.

    MATH  MathSciNet  Google Scholar 

  98. R.P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann., 278 (1987), 219–271.

    Article  MATH  MathSciNet  Google Scholar 

  99. R.P. Langlands and D. Shelstad, Descent for transfer factors, In: The Grothendieck Festschrift, Vol. II, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990, pp. 485–563.

  100. G. Laumon and B.C. Ngô, Le lemme fondamental pour les groupes unitaires, Ann. of Math. (2), 168 (2008), 477–573.

    Google Scholar 

  101. B. Mazur, Deforming Galois representations, In: Galois Groups Over \({\mathbb{Q}}\) , (eds. Y. Ihara, K. Ribet and J.-P. Serre), Springer-Verlag, 1989, pp. 385–437.

  102. B. Mazur, Finding meaning in error terms, Bull. Amer. Math. Soc., 45 (2008), 185–228.

    Article  MATH  MathSciNet  Google Scholar 

  103. J.S. Milne, Introduction to Shimura varieties, In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, (eds. J. Arthur, D. Ellwood and R. Kottwitz), Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005, pp. 265–378.

  104. A. Minguez, Unramified representations of unitary groups, chapter in [Book1].

  105. C. Mœglin, Classification et changement de base pour les séries discrètes des groupes unitaires p-adiques, Pacific J. Math., 233 (2007), 159–204.

    Google Scholar 

  106. C. Mœglin, Paquets d’Arthur discrets pour un groupe classique p-adique, In: Automorphic Forms and L-functions II. Local Aspects, Proceedings of the Conference in Honor of S. Gelbart, (eds. D. Ginzburg, E. Lapid and D. Soudry), Contemp. Math., 489, Amer. Math. Soc., 2009, pp. 179–258.

  107. C. Mœglin, Multiplicité 1 dans les paquets d’Arthur, In: Proceedings of the Conference in Honor of F. Shahidi, to appear.

  108. S. Morel, Étude de la Cohomologie de Certaines Variétés de Shimura Non Compactes, Ann. of Math. Stud., in press.

  109. L. Moret-Bailly, Groupes de Picard et problèmes de Skolem. II, Ann. Sci. École Norm. Sup. (4), 22 (1989), 181–194.

  110. B.C. Ngô, Fibration de Hitchin et endoscopie, Invent. Math., 164 (2006), 399–453.

  111. B.C. Ngô, Le lemme fondamental pour les algèbres de Lie, in press.

  112. B.C. Ngô, Decomposition theorem and abelian fibration, to appear in [Book1].

  113. R. Ramakrishna, Infinitely ramified Galois representations, Ann. of Math. (2), 151 (2000), 793–815.

  114. D. Ramakrishnan, Modularity of the Rankin–Selberg L-series, and multiplicity one for SL(2), Ann. of Math. (2), 152 (2000), 45–111.

  115. M. Rapoport, A guide to the reduction modulo p of Shimura varieties, In: Automorphic Forms. I, Astérisque, 298, Soc. Math. France, 2005, pp. 271–318.

  116. M. Rapoport and T. Zink, Period Spaces for p-divisible Groups, Ann. of Math. Stud., 141, Princeton Univ. Press, 1996.

  117. J.D. Rogawski, Automorphic Representations of Unitary Groups in Three Variables, Ann. of Math. Stud., 123, Princeton Univ. Press, 1990.

  118. J.-P. Serre, Abelian ℓ-adic Representations and Elliptic Curves, Benjamin, New York, 1968.

  119. J.-P. Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations ℓ-adiques, In: Motives, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994, pp. 377–400.

  120. F. Shahidi, On certain L-functions, Amer. J. Math., 103 (1981), 297–355.

    Article  MATH  MathSciNet  Google Scholar 

  121. D. Shelstad, Characters and inner forms of a quasi-split group over \({\mathbb R}\) Compositio Math., 39 (1979), 11–45.

  122. D. Shelstad, L-indistinguishability for real groups, Math. Ann., 259 (1982), 385–430.

    Article  MATH  MathSciNet  Google Scholar 

  123. G. Shimura, Moduli of abelian varieties and number theory, In: Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., 9, Amer. Math. Soc., Providence, RI, 1966, pp. 312–332.

  124. G. Shimura, On arithmetic automorphic functions. In: Actes du Congrès International des Mathématiciens, Nice, 1970, Tome 2, pp. 343–348.

  125. G. Shimura and Y. Taniyama, Complex Multiplication of Abelian Varieties and its Applications to Number Theory, Publ. Math. Soc. Japan, 6, Math. Soc. Japan, Tokyo, 1961.

  126. S.-W. Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2), in press.

  127. C. Skinner and A. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math., 89 (1999), 5–126.

    Google Scholar 

  128. A. Snowden and A. Wiles, Bigness in compatible systems, manuscript, 2008.

  129. C.M. Sorensen, A patching lemma, to appear in [Book2].

  130. R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math., 98 (1989), 265–280.

    Article  MATH  MathSciNet  Google Scholar 

  131. R. Taylor, Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu, 1 (2002), 125–143.

    Google Scholar 

  132. R. Taylor, Galois representations, In: Proceedings of ICM 2002, Vol. I, Higher Ed. Press, pp. 449–474.

  133. R. Taylor, On the meromorphic continuation of degree two L-functions, Doc. Math., 2006 (2006), Extra vol., 729–779.

  134. R. Taylor, Automorphy for some ℓ-adic lifts of automorphic mod ℓ Galois representations. II, Publ. Math. Inst. Hautes Études Sci., 108 (2008), 183–239.

  135. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2), 141 (1995), 553–572.

    Article  MATH  MathSciNet  Google Scholar 

  136. J.-L. Waldspurger, Le lemme fondamental implique le transfert, Compositio Math., 105 (1997), 153–236.

    Article  MATH  MathSciNet  Google Scholar 

  137. J.-L. Waldspurger, Intégrales Orbitales Nilpotentes et Endoscopie pour les Groupes Classiques Non Ramifiés, Astérisque, 269, Soc. Math. France, 2001.

  138. J.-L. Waldspurger, Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu, 5 (2006), 423–525.

    Article  MATH  MathSciNet  Google Scholar 

  139. J.-L. Waldspurger, L’endoscopie Tordue N’est Pas si Tordue, Mem. Amer. Math. Soc., 194, Amer. Math. Soc., 2008.

  140. J.-L. Waldspurger, Endoscopie et changement de caractéristique: intégrales orbitales pondérées, Ann. Inst. Fourier, 59 (2009), 1753–1818.

    Google Scholar 

  141. J.-L. Waldspurger, A propos du lemme fondamental tordu, chapter in [Book1].

  142. A. Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math., 94 (1988), 529–573.

    Google Scholar 

  143. A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2), 141 (1995), 443–551.

    Article  MATH  MathSciNet  Google Scholar 

  144. H. Yoshida, On an analogue of the Sato conjecture, Invent. Math. 19 (1973), 261–277.

    Article  MATH  MathSciNet  Google Scholar 

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  1. UFR de Mathématiques, Université Paris 7, 2 Pl. Jussieu, 75251, Paris cedex 05, France

    Michael Harris

  2. Institut des Mathématiques de Jussieu, U.M.R. 7586 du CNRS. Membre, Institut Universitaire de France, Paris cedex 05, France

    Michael Harris

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Communicated by: Takeshi Saito

This article is based on the 7th Takagi Lectures that the author delivered at the University of Tokyo on November 21–23, 2009.

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Harris, M. Arithmetic applications of the Langlands program. Jpn. J. Math. 5, 1–71 (2010). https://doi.org/10.1007/s11537-010-0945-6

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