Around the Langlands Program

Jahresbericht der Deutschen Mathematiker-Vereinigung - Tập 120 Số 1 - Trang 3-40 - 2018
Anne-Marie Aubert1
1Institut de Mathématiques de Jussieu – Paris Rive Gauche, U.M.R. 7586 du C.N.R.S., U.P.M.C., Paris, France

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Tài liệu tham khảo

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

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

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

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

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)

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

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

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)

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)

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)

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)

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

Aubert, A.-M., Baum, P., Plymen, R.J., Solleveld, M.: Conjectures about p $p$ -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)

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

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

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

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

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

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

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

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

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

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)

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fargues, L., Fontaine, J.-M.: Vector bundles on curves and p $p$ -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)

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

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

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

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

Goldfeld, D., Hundley, J.: Automorphic Representations and L $L$ -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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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)

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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)

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

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

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

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

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

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

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

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)

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

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

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)

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

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

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