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Around the Langlands Program

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Notes

  1. A representation of \(G\) of finite dimension \(n\) consists in attaching to each element \(g\) in \(G\) an invertible square matrix \(M_{g}\) of size \(n\) that satisfies \(M_{g\cdot g'}=M_{g} M_{g'}\) for all \(g\) and \(g'\) in \(G\).

  2. An automorphism \(\sigma \) of a field \(L\) is a bijective application \(\sigma \colon L \to L\) such that \(\sigma (a+b)=\sigma (a)+ \sigma (b)\) and \(\sigma (ab)=\sigma (a)\sigma (b)\), for all \(a\) and \(b\) in \(L\). The set of automorphisms of \(L\), provided with composition law of applications, forms a group; the automorphisms of \(L\) that leave unchanged the elements of \(K\) is a subgroup of the latter.

  3. A field extension \(L/K\) is called algebraic if every element of \(L\) is algebraic over \(K\), i.e., if every element of \(L\) is a root of some non-zero polynomial with coefficients in \(K\). A field \(L\) is called algebraically closed if it contains a root for every non-constant polynomial in \(L[X]\), the ring of polynomials in the variable \(X\) with coefficients in \(L\). An algebraic closure of \(K\) is an algebraic extension of \(K\) that is algebraically closed.

  4. The group \(\mathrm{GL}_{n}({\mathbb{A}}_{{\mathbb{Q}}})\) consists in invertible \(n\times n\) matrices with entries in \({\mathbb{A}}_{{\mathbb{Q}}}\).

  5. More precisely, these are representations of a certain group \({\mathcal{L}}_{{\mathbb{Q}}}\), called the Langlands group of ℚ, which is close to the group \(\varGamma_{{\mathbb{Q}}}\). However, the group \({\mathcal{L}} _{{\mathbb{Q}}}\) is complicated and its existence is still conjectural in general, even, in the case when \(n=2\), see Sect. 10.

  6. I.e., \(K\) is either a finite extension of ℚ or a finite extension of \({\mathbb{F}}_{q}(t)\), where \({{\mathbb{F}} _{q}}\) is a finite field with \(q\) elements and \(t\) is an indeterminate. In the first case, \(K\) is called a number field, otherwise it is called a global function field.

  7. See Sect. 7.

  8. Among modular forms, the cusp forms are distinguished by the vanishing in their Fourier series expansion \(\sum_{n\ge 0}a_{n} q^{n}\) of the constant coefficient \(a_{0}\).

  9. See Sect. 3.

  10. See Sect. 3.

  11. Any inner form of \(\mathrm{GL}_{n}(F)\) is given by the group of invertible \(m\times m\) matrices \(\mathrm{GL}_{m}(D)\), with entries in a division algebra \(D\) with center \(F\) and such \(\dim_{F} (D) = d^{2}\), where \(dm=n\).

  12. A tempered representation of a linear semisimple Lie group \(G\) is a representation that has a basis whose matrix coefficients lie in the \(L^{p}\) space \(L^{2+\varepsilon }(G)\), for any \(\varepsilon > 0\). This condition is slightly weaker than the condition that the matrix coefficients are square-integrable, in other words lie in \(L^{2}(G)\), which would be the definition of a discrete series representation.

  13. See Sect. 3.

  14. See Equation (6) in Sect. 4.

  15. See Sect. 5.

  16. I.e., an \(L\) -homomorphism as defined in 5.1.

  17. I.e., if any irreducible polynomial with coefficients in \(F\) and a root in the extension actually splits into a product of linear factors as polynomial with coefficients in the extension.

  18. A Hausdorff topological space is a topological space in which distinct points have disjoint neighborhoods.

  19. This means that it is possible to consider the \(F\)-points of \({\mathbf{G}}\).

  20. It is a direct product if \(G\) is \(F\)-split.

  21. A distribution on \(G\) is defined to be a linear functional on \(C_{{\mathrm{c}}}^{\infty }(G)\). A distribution \(D\) on \(G\) is said to be \(G\)-invariant (or invariant) if \(D(fg) = D(f)\) for all \(g\in G\).

  22. A proof for \(G=\mathrm{GL}_{2}(F)\) can be found in [31, Prop. 9.4 and Cor. 10.2]; for \(G\) an arbitrary reductive \(p\)-adic group, it is proved in [88, Theorem VI.2.2].

  23. Recall that two elements \(\gamma \) and \(\gamma '\) in \(G={\mathbf{G}}(F)\) are said to be conjugate if there is \(g\in G\) such that \(\gamma '=g\gamma g ^{-1}\). A conjugacy class \(C\) is the set of conjugates of an element in \(G\).

  24. An element \(\gamma \) of \({\mathbf{G}}\) is called semisimple if it is contained in a torus, it is called regular if the identity component of its centralizer is a maximal torus, and strongly regular if its centralizer itself is a maximal torus. A regular semisimple element \(\gamma \) of \(G\) is called elliptic if its centralizer in \(G\) is compact modulo the center of \(G\). In \(G=\mathrm{GL}_{n}(F)\), the semi-simple conjugacy classes \((\gamma )\) are classified by their characteristic polynomial \(P_{\gamma }(X)\in F[X]\); the element \(\gamma \) is regular if and only if the roots of \(P_{\gamma }(X)\) are simple, and \(\gamma \) is elliptic if the quotient \(F[X]/P_{\gamma }(X)F[X]\) is a field of degree \(n\) over \(F\).

  25. A discrete series representation of a locally compact topological group \(G\) is an irreducible unitary representation of \(G\) that is a subrepresentation of the left regular representation of \(G\) on \(L^{^{2}}(G)\).

  26. Note that \(f_{\pi }\) is highly non-unique. However as regards invariant harmonic analysis this plays no role.

  27. This reflects the fact that we consider simultaneously all the inner twists of a given group \(G\).

  28. The set of (equivalence classes of) inner twists of \({\mathbf{G}}\) is parametrized by the Galois cohomology group \(H^{1}(F,{\mathbf{G}}^{*} _{\mathrm{ad}})\); we parametrize the quasi-split twist of \({\mathbf{G}}\) by \(\zeta =1\).

  29. The space \(L^{2}({\mathbf{G}}(K) \backslash {\mathbf{G}}({\mathbb{A}}_{K}), \omega )\) is an Hilbert space under the inner product \((f_{1},f_{2})\mapsto \int_{{\mathbf{Z}}({\mathbb{A}}_{K}){\mathbf{G}}(K)\setminus {\mathbf{G}}( {\mathbb{A}}_{K})} f_{1}(g)\overline{f_{2}(g)}dg\), and it admits a unitary action of \({\mathbf{G}}({\mathbb{A}}_{K})\) by right translations: \({(g' \cdot f)}(g):= f(gg')\) for \(g'\in {\mathbf{G}}( {\mathbb{A}}_{K})\).

  30. The modular group is the group of transformations \(\varGamma (1):=\mathrm{GL}_{2}({\mathbb{Z}})^{+}\) considered in Sect. 7. It is isomorphic to the projective special linear group \(\mathrm{PSL}_{2}({\mathbb{Z}})\) of \(2 \times 2\) matrices with coefficients in ℤ and unit determinant.

  31. See for instance [25, §1.6].

  32. We have \(G=\mathrm{Res}_{L/K}(\mathrm{GL}_{n})\), where \(\mathrm{Res}_{L/K}\) is the Weil restriction of scalars defined in [99].

  33. In the case of Example 9.1, \({\mathbf{H}}\) corresponds to a twisted endoscopic group for \({\mathbf{G}}\). The theory of twisted endoscopy is a key ingredient in [7].

  34. An \(E\)-Banach space is a topological \(E\)-vector space which is complete with respect to the topology given by a norm; when \(E={\overline{\mathbb{Q}}_{\ell }}\), it is called an \(\ell \)-adic Banach space.

  35. A Größen-character—or Hecke character—of \(K\) is a family of morphisms \(\chi =(\chi _{x})\) indexed by the places of \(K\), with \(\chi_{x}\colon K_{x}^{ \times }\), satisfying \(\prod_{x\in |C|}\chi_{x}(a)=1\) for each \(a\in K^{\times }\).

  36. A function is called smooth if it is invariant by an open compact subgroup.

  37. It is the group of isomorphism classes of invertible sheaves (or line bundles) on \(C\), with the group operation being tensor product.

References

  1. Abe, N., Herzig, F., Henniart, G., Vignéras, M.-F.: A classification of irreducible admissible mod \(p\) representations of \(p\)-adic reductive groups. J. Am. Math. Soc. 30(2), 495–559 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arthur, J.: A Paley-Wiener theorem for real reductive groups. Acta Math. 150, 1–89 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Arthur, J.: A note of the automorphic Langlands group. Can. Math. Bull. 45(4), 466–482 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arthur, J.: The principle of functoriality. In: Mathematical Challenges of the 21st Century, Los Angeles, CA, 2000. Bull. Amer. Math. Soc., vol. 40, pp. 39–53 (2003)

    Google Scholar 

  6. Arthur, J.: A note on L-packets. Pure Appl. Math. Q. 2(1), 199–217 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arthur, J.: The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups. Colloquium Publications, vol. 61. Am. Math. Soc., Providence (2013)

    MATH  Google Scholar 

  8. Arthur, J., Clozel, L.: Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. Annals of Mathematics Studies, vol. 120. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  9. Aubert, A.-M., Baum, P., Plymen, R.J., Solleveld, M.: Geometric structure in smooth dual and local Langlands correspondence. Jpn. J. Math. 9, 99–136 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Aubert, A.-M., Baum, P., Plymen, R.J., Solleveld, M.: On the local Langlands correspondence for non-tempered representations. Münster J. Math. 7, 27–50 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Aubert, A.-M., Baum, P., Plymen, R.J., Solleveld, M.: Depth and the local Langlands correspondence. In: Arbeitstagung Bonn. Progress in Math., vol. 2013. Birkhäuser, Basel (2016)

    Google Scholar 

  12. Aubert, A.-M., Baum, P., Plymen, R.J., Solleveld, M.: The local Langlands correspondence for inner forms of \(\mathrm{SL} _{n}\). Res. Math. Sci. 3, 32 (2016)

    Article  MATH  Google Scholar 

  13. Aubert, A.-M., Baum, P., Plymen, R.J., Solleveld, M.: Conjectures about \(p\)-adic groups and their noncommutative geometry. In: Proceedings of the Conference Around Langlands Correspondences. Contemp. Math. Amer. Math. Soc., vol. 691, pp. 15–51 (2017)

    Chapter  Google Scholar 

  14. Aubert, A.-M., Mendes, S., Plymen, R., Solleveld, M.: On \(L\)-packets and depth for \(\mathrm{SL} _{2}(K)\) and its inner form. Int. J. Number Theory (2017). doi:10.1142/S1793042117501421

    MATH  Google Scholar 

  15. Aubert, A.-M., Moussaoui, A., Solleveld, M.: Generalizations of the Springer correspondence and cuspidal Langlands parameters. Manuscripta Math. arXiv:1511.05335, in press

  16. Aubert, A.-M., Moussaoui, A., Solleveld, M.: Affine Hecke algebras for Langlands parameters. arXiv:1701.03593

  17. Baker, A.: Matrix Groups. An Introduction to Lie Group Theory. Springer Undergraduate Mathematics Series. Springer, London (2002)

    MATH  Google Scholar 

  18. Barthel, L., Livné, R.: Irreducible modular representations of \(\mathrm{GL} _{2}\) of a local field. Duke Math. J. 55, 261–292 (1994)

    Article  MATH  Google Scholar 

  19. Barthel, L., Livné, R.: Modular representations of \(\mathrm{GL} _{2}\) of a local field: the ordinary unramified case. J. Number Theory 55, 1–27 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Berger, L.: Représentations modulaires de \(\mathrm{GL} _{2}({\mathbb{Q} _{p}})\) et représentations galoisiennes de dimension 2. Astérisque 330, 265–281 (2010)

    Google Scholar 

  21. Berger, L.: La correspondance de Langlands locale \(p\)-adique pour \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\). In: Bourbaki Seminar (2010)

    Google Scholar 

  22. Berger, L., Breuil, C.: Sur quelques représentations potentiellement cristallines de \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\). Astérisque 330, 155–211 (2010)

    MATH  Google Scholar 

  23. Bernstein, J., Gelbart, S.: Bump, D., Cogdell, J., de Shalit, E., Gaitsgory, D., Kowalski, E., Kudla, S. (eds.) An Introduction to the Langlands Program. Birkhäuser, Boston (2004)

    Google Scholar 

  24. Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Math., vol. 126. Springer, New York (1991)

    Book  MATH  Google Scholar 

  25. Borel, A.: Lie Groups and Linear Algebraic Groups. I. Complex and Real Groups. Lie Groups and Automorphic Forms. AMS/IP Stud. Adv. Math., vol. 37, pp. 1–49. Am. Math. Soc., Providence (2006)

    MATH  Google Scholar 

  26. Breuil, C.: Invariant ℒ et série spéciale \(p\)-adique. Ann. Sci. Éc. Norm. Supér. 37, 559–610 (2004)

    Article  MATH  Google Scholar 

  27. Breuil, C.: Introduction générale. In: Représentations \(p\)-adiques de groupes \(p\)-adiques. I. Représentations galoisiennes et \((\varphi,\varGamma)\)-modules. Astérisque, vol. 319 pp. 1–12 (2008)

    Google Scholar 

  28. Breuil, C.: The emerging \(p\)-adic Langlands programme. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 203–230. Hindustan Book Agency, New Delhi (2010)

    Google Scholar 

  29. Brumley, F., Gomez Aparicio, M., Minguez, A.: Around Langlands correspondences. Contemp. Math. Am. Math. Soc. 691, 376 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Bump, D.: Automorphic Forms and Representations. Cambridge Studies in Adv. Math., vol. 55. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  31. Bushnell, C., Henniart, G.: The Local Langlands Conjecture for \(\mathrm{GL} (2)\). Grundlehren der math. Wissenschaften, vol. 335. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  32. Carayol, H.: Preuve de la conjecture de Langlands locale pour \(\mathrm{GL} _{n}\): travaux de Harris-Taylor et Henniart. In: Séminaire Bourbaki, Exp. no. 857. Astérisque, vol. 266, 191–243 (2000)

    Google Scholar 

  33. Cogdell, J., Kim, H., Murty, M.R.: Lectures on Automorphic \(L\)-Functions. Fields Institute Monographs, vol. 20. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

  34. Colmez, P.: La série principale unitaire de \(\mathrm{GL} _{2}({\mathbb{Q} _{p}})\). Astérisque 330, 213–262 (2010)

    Google Scholar 

  35. Colmez, P.: \((\varphi ,\varGamma )\)-modules et représentations du mirabolique de \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\). Astérisque 330, 61–153 (2010)

    Google Scholar 

  36. Colmez, P.: Représentations de \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\) et “\(( \varphi ,\varGamma )\)-modules. Astérisque 330, 281–509 (2010)

    Google Scholar 

  37. Colmez, P.: Le programme de Langlands \(p\)-adique. In: European Congress of Mathematics, pp. 259–284. Eur. Math. Soc., Zürich (2013)

    Google Scholar 

  38. Delorme, P.: Multipliers for the convolution algebra of left and right \(K\)-finite compactly supported smooth functions on a semisimple Lie group. Invent. Math. 75, 9–23 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Diamond, F., Shurman, J.: A First Course in Modular Forms. Grad. Texts in Math. Springer, Berlin (2010)

    MATH  Google Scholar 

  40. Drinfeld, V.: Proof of the global Langlands conjecture for \(\mathrm{GL} (2)\) over a function field. Funct. Anal. Appl. 11, 223–225 (1977)

    Article  Google Scholar 

  41. Fargues, L.: Geometrization of the local Langlands correspondence: an overview. arXiv:1602.00999

  42. Fargues, L., Fontaine, J.-M.: Vector bundles on curves and \(p\)-adic Hodge theory. In: Automorphic forms and Galois Representations. London Math. Soc. Lecture Note Ser., vol. 415, pp. 17–104. Cambridge University Press, Cambridge (2014)

    Chapter  Google Scholar 

  43. Frenkel, E.: Recent advances in the Langlands program. Bull. Am. Math. Soc. 41(2), 151–184 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  44. Frenkel, E.: Langlands program, trace formulas, and their geometrization. Bull. Am. Math. Soc. 50(1), 1–55 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gaitsgory, D.: Informal introduction to geometric Langlands. In: An Introduction to the Langlands Program, Jerusalem, 2001, pp. 269–281. Birkhäuser, Boston (2003)

    Google Scholar 

  46. Gelbart, S.: An elementary introduction to the Langlands program. Bull. Am. Math. Soc. 10(2), 177–219 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  47. Goldfeld, D., Hundley, J.: Automorphic Representations and \(L\)-Functions for the General Linear Group, vols. I and II. Cambridge Studies in Advanced Mathematics, vols. 129, 130. Cambridge University Press, Cambridge (2011). With exercises and a preface by Xander Faber

    Book  MATH  Google Scholar 

  48. Harris, M.: Arithmetic applications of the Langlands program. Jpn. J. Math. 5(1), 1–71 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Princeton University Press, Princeton (2001)

    MATH  Google Scholar 

  50. Heiermann, V.: Paramètres de Langlands et Algèbres d’entrelacement. Int. Math. Res. Not. 9, 1607–1623 (2010)

    MATH  Google Scholar 

  51. Heiermann, V.: Local Langlands correspondence for classical groups and affine Hecke algebras (2015). arXiv:1502.04357

  52. Heiermann, V.: Opérateurs d’entrelacement et algèbres de Hecke avec paramètres d’un groupe réductif \(p\)-adique—le cas des groupes classiques. Sel. Math. 17(3), 713–756 (2011)

    Article  MATH  Google Scholar 

  53. Henniart, G.: Caractérisation de la correspondance de Langlands locale par les facteurs \(\epsilon \) de paires. Invent. Math. 113, 339–350 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Hiraga, K., Saito, H.: On L-Packets for Inner Forms of \(\mathrm{SL} _{n}\). Mem. Am. Math. Soc. 1013, vol. 215 (2012)

    MATH  Google Scholar 

  56. Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Math., vol. 21. Springer, New York (1975)

    MATH  Google Scholar 

  57. Kaletha, T., Minguez, A., Shin, S.W., White, P.-J.: Endoscopic classification of representations: Inner forms of unitary groups. Preprint (2014)

  58. Kazhdan, D.: Cuspidal geometry of \(p\)-adic groups. J. Anal. Math. 47, 1–36 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  59. Khare, C.: On the local Langlands correspondence \(\operatorname{mod} \ell \). J. Number Theory 88, 357–365 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  60. Knapp, A.W.: Introduction to the Langlands program. In: Representation Theory and Automorphic Forms, Edinburgh, 1996. Proc. Sym. Pure Math., vol. 61. Am. Math. Soc., Providence (1997)

    Google Scholar 

  61. Kottwitz, R., Shelstad, D.: Foundations of twisted endoscopy. Astérisque 255, 1–190 (1999)

    MathSciNet  MATH  Google Scholar 

  62. Kowalski, E.: An Introduction to the Representation Theory of Groups. Graduate Studies in Mathematics, vol. 155. Am. Math. Soc., Providence (2014)

    MATH  Google Scholar 

  63. Kudla, S.: From modular forms to automorphic representations. In: Bernstein, J., Gelbart, S. (eds.) An Introduction to the Langlands Program, pp. 133–151. Birkhäuser, Boston (2003)

    Google Scholar 

  64. Lafforgue, L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147, 1–241 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  65. Lafforgue, V.: K-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes. Invent. Math. 149(1), 1–95 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  66. Lafforgue, V.: Introduction to chtoucas for reductive groups and to the global Langlands parameterization. arXiv:1404.6416

  67. Lafforgue, V.: Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale. arXiv:1209.5352

  68. Langlands, R.P.: Letter to André Weil (1967). http://sunsite.ubc.ca/DigitalMathArchive/Langlands/functoriality.html

    Google Scholar 

  69. Langlands, R.P.: Problems in the theory of automorphic forms. In: Lectures in Modern Analysis and Applications. Lecture Notes in Math., vol. 170, pp. 18–61. Springer, New York (1970)

    Google Scholar 

  70. Langlands, R.P.: On the Classification of Irreducible Representations of Real Algebraic Groups. AMS, Providence (1973)

    MATH  Google Scholar 

  71. Langlands, R.P.: On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Math., vol. 544. Springer, Berlin (1976)

    MATH  Google Scholar 

  72. Langlands, R.P.: Automorphic representations, Shimura varieties, and motives. Ein Märchen. In: Automorphic Forms, Representations and \(L\)-Functions, Oregon State Univ., Corvallis, Ore., 1977. Proc. Sympos. Pure Math., vol. 33, pp. 205–246. Am. Math. Soc., Providence (1979). Part 2

    Chapter  Google Scholar 

  73. Langlands, R.P.: Les débuts d’une formule des traces stables. Publ. math. de l’Université Paris, vol. VII (1983)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  75. Laumon, G.: Chtoucas de Drinfeld et correspondance de Langlands. Gaz. Math. 88, 11–33 (2001)

    MATH  Google Scholar 

  76. Laumon, G.: La correspondance de Langlands sur les corps de fonctions (d’après Laurent Lafforgue) In: Séminaire Bourbaki Vol. 1999/2000. Astérisque, vol. 276, pp. 207–265 (2002). Exp. No. 973

    Google Scholar 

  77. Laumon, G.: Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld- Langland. In: Séminaire Bourbaki Vol. 2001/2002. Astérisque, vol. 290, pp. 267–284 (2003). Exp. No. 906, ix

    Google Scholar 

  78. Laumon, G., Rapoport, M., Stuhler, U.: Elliptic sheaves and the Langlands correspondence. Invent. Math. 113, 217–338 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  79. Lusztig, G.: Some examples of square-integrable representations of semisimple \(p\)-adic groups. Trans. Am. Math. Soc. 277, 623–653 (1983)

    MathSciNet  MATH  Google Scholar 

  80. Lusztig, G.: Intersection cohomology complexes on a reductive group. Invent. Math. 75(2), 205–272 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  81. Mœglin, C., Waldspurger, J.-L.: Spectral Decomposition and Eisenstein Series. Une paraphrase de l’Écriture. Cambridge Tracts in Mathematics, vol. 113. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  82. Mœglin, C., Waldspurger, J.-L.: Stabilisation de la formule des traces tordue. Progress in Math., vol. 316–317. Birkhäuser, Boston (2017)

    MATH  Google Scholar 

  83. Mok, C.P.: Endoscopic Classification of Representations of Quasi-Split Unitary Groups. Mem. Amer. Math. Soc., vol. 235 (2015), no. 1108, vi+248 pp.

    MATH  Google Scholar 

  84. Moussaoui, A.: Centre de Bernstein enrichi des groupes classiques. Representation Theory, in press

  85. Murty, M.R.: A motivated introduction to the Langlands program. In: Advances in Number Theory, Kingston, ON, 1991, pp. 37–66. Oxford University Press, Oxford (1993)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  87. Ngô, B.C.: Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111, 1–169 (2010)

    Article  MATH  Google Scholar 

  88. Renard, D.: Représentations des groupes réductifs \(p\)-adiques, Cours spécialisés. Société Mathématique de France, vol. 17 (2010)

    MATH  Google Scholar 

  89. Scholze, P.: The local Langlands correspondence for \(\mathrm{GL} _{n}\) over \(p\)-adic fields. Invent. Math. 192, 663–715 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  90. Scholze, P.: Perfectoid spaces: a survey. In: Current Developments in Mathematics 2012, pp. 193–227. International Press, Somerville (2013)

    Google Scholar 

  91. Serre, J.-P.: Cours d’arithmétique. P.U.F, Paris (1970)

    MATH  Google Scholar 

  92. Springer, T.A.: Linear Algebraic Groups, 2nd edn. Progress in Math., vol. 9. Birkhäuser, Boston (1998)

    Book  MATH  Google Scholar 

  93. Tate, J.: Fourier analysis in number fields and Hecke’s zeta-functions (Ph.D. thesis, 1950). In: Cassels, J., Frölich, A. (eds.) Algebraic Number Theory, pp. 305–347. Thompson, Washington (1950)

    Google Scholar 

  94. Vignéras, M.-F.: Représentations \(\ell \)-modulaires d’un groupe réductif \(p\)-adique avec \(\ell \ne p\). Progr. Math., vol. 137. Birkhäuser, Boston (1996)

    Google Scholar 

  95. Vignéras, M.-F.: Correspondance de Langlands semi-simple pour \(\mathrm{GL} (n,F)\) modulo \(\ell \ne p\). Invent. Math. 144, 177–223 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  96. Vogan, D.: The local Langlands conjecture. In: Representation Theory of Groups and Algebras. Contemp. Math., vol. 145, pp. 305–379. Am. Math. Soc., Providence (1993)

    Chapter  Google Scholar 

  97. Waldspurger, J.-L.: Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondammental. Can. J. Math. 43, 852–896 (1991)

    Article  MATH  Google Scholar 

  98. Waldspurger, J.-L.: L’endoscopie tordue n’est pas si tordue. Mem. Amer. Math. Soc., vol. 194 (2008)

    MATH  Google Scholar 

  99. Weil, A.: Adeles and Algebraic Groups. Progress in Math., vol. 23. Birkhäuser, Basel (1982)

    Book  MATH  Google Scholar 

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  1. Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C., 4 place Jussieu, 75005, Paris, France

    Anne-Marie Aubert

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Aubert, AM. Around the Langlands Program. Jahresber. Dtsch. Math. Ver. 120, 3–40 (2018). https://doi.org/10.1365/s13291-017-0171-8

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