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Jahresbericht der Deutschen Mathematiker-Vereinigung
Erschienen in: Jahresbericht der Deutschen Mathematiker-Vereinigung 1/2018

13.09.2017 | Survey Article

Around the Langlands Program

verfasst von: Anne-Marie Aubert

Erschienen in: Jahresbericht der Deutschen Mathematiker-Vereinigung | Ausgabe 1/2018

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Robert P. Langlands is a Canadian mathematician. He became professor at the Institute for Advanced Study (IAS) of Princeton in 1972, and has received numerous awards, among them the Wolf prize, in 1996, the Nemmers prize, in 2006, and the Shaw prize, in 2007. …
Vorheriger Artikel Preface
Nächster Artikel David Hilbert: Die Theorie der algebraischen Zahlkörper

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Fußnoten
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.
 
Literatur
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) MathSciNetCrossRefMATH
2.
3.
4.
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)
6.
7.
Arthur, J.: The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups. Colloquium Publications, vol. 61. Am. Math. Soc., Providence (2013) MATH
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
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) MathSciNetCrossRefMATH
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) MathSciNetMATH
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)
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) CrossRefMATH
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) CrossRef
14.
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.
17.
Baker, A.: Matrix Groups. An Introduction to Lie Group Theory. Springer Undergraduate Mathematics Series. Springer, London (2002) MATH
18.
Barthel, L., Livné, R.: Irreducible modular representations of \(\mathrm{GL} _{2}\) of a local field. Duke Math. J. 55, 261–292 (1994) CrossRefMATH
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) MathSciNetCrossRefMATH
20.
Berger, L.: Représentations modulaires de \(\mathrm{GL} _{2}({\mathbb{Q} _{p}})\) et représentations galoisiennes de dimension 2. Astérisque 330, 265–281 (2010)
21.
Berger, L.: La correspondance de Langlands locale \(p\)-adique pour \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\). In: Bourbaki Seminar (2010)
22.
Berger, L., Breuil, C.: Sur quelques représentations potentiellement cristallines de \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\). Astérisque 330, 155–211 (2010) MATH
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)
24.
Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Math., vol. 126. Springer, New York (1991) CrossRefMATH
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
26.
Breuil, C.: Invariant ℒ et série spéciale \(p\)-adique. Ann. Sci. Éc. Norm. Supér. 37, 559–610 (2004) CrossRefMATH
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)
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)
29.
Brumley, F., Gomez Aparicio, M., Minguez, A.: Around Langlands correspondences. Contemp. Math. Am. Math. Soc. 691, 376 (2017) MathSciNetMATH
30.
Bump, D.: Automorphic Forms and Representations. Cambridge Studies in Adv. Math., vol. 55. Cambridge University Press, Cambridge (1997) CrossRefMATH
31.
Bushnell, C., Henniart, G.: The Local Langlands Conjecture for \(\mathrm{GL} (2)\). Grundlehren der math. Wissenschaften, vol. 335. Springer, Berlin (2006) CrossRefMATH
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)
33.
Cogdell, J., Kim, H., Murty, M.R.: Lectures on Automorphic \(L\)-Functions. Fields Institute Monographs, vol. 20. American Mathematical Society, Providence (2004) MATH
34.
Colmez, P.: La série principale unitaire de \(\mathrm{GL} _{2}({\mathbb{Q} _{p}})\). Astérisque 330, 213–262 (2010)
35.
Colmez, P.: \((\varphi ,\varGamma )\)-modules et représentations du mirabolique de \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\). Astérisque 330, 61–153 (2010)
36.
Colmez, P.: Représentations de \(\mathrm{GL} _{2}({\mathbb{Q}_{p}})\) et “\(( \varphi ,\varGamma )\)-modules. Astérisque 330, 281–509 (2010)
37.
Colmez, P.: Le programme de Langlands \(p\)-adique. In: European Congress of Mathematics, pp. 259–284. Eur. Math. Soc., Zürich (2013)
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) MathSciNetCrossRefMATH
39.
Diamond, F., Shurman, J.: A First Course in Modular Forms. Grad. Texts in Math. Springer, Berlin (2010) MATH
40.
Drinfeld, V.: Proof of the global Langlands conjecture for \(\mathrm{GL} (2)\) over a function field. Funct. Anal. Appl. 11, 223–225 (1977) CrossRef
41.
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) CrossRef
43.
44.
45.
Gaitsgory, D.: Informal introduction to geometric Langlands. In: An Introduction to the Langlands Program, Jerusalem, 2001, pp. 269–281. Birkhäuser, Boston (2003)
46.
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 CrossRefMATH
48.
49.
Harris, M., Taylor, R.: The Geometry and Cohomology of Some Simple Shimura Varieties. Princeton University Press, Princeton (2001) MATH
50.
Heiermann, V.: Paramètres de Langlands et Algèbres d’entrelacement. Int. Math. Res. Not. 9, 1607–1623 (2010) MATH
51.
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) CrossRefMATH
53.
Henniart, G.: Caractérisation de la correspondance de Langlands locale par les facteurs \(\epsilon \) de paires. Invent. Math. 113, 339–350 (1993) MathSciNetCrossRefMATH
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) MathSciNetCrossRefMATH
55.
Hiraga, K., Saito, H.: On L-Packets for Inner Forms of \(\mathrm{SL} _{n}\). Mem. Am. Math. Soc. 1013, vol. 215 (2012) MATH
56.
Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Math., vol. 21. Springer, New York (1975) MATH
57.
Kaletha, T., Minguez, A., Shin, S.W., White, P.-J.: Endoscopic classification of representations: Inner forms of unitary groups. Preprint (2014)
58.
59.
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)
61.
Kottwitz, R., Shelstad, D.: Foundations of twisted endoscopy. Astérisque 255, 1–190 (1999) MathSciNetMATH
62.
Kowalski, E.: An Introduction to the Representation Theory of Groups. Graduate Studies in Mathematics, vol. 155. Am. Math. Soc., Providence (2014) MATH
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)
64.
65.
Lafforgue, V.: K-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes. Invent. Math. 149(1), 1–95 (2002) MathSciNetCrossRefMATH
66.
67.
68.
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)
70.
Langlands, R.P.: On the Classification of Irreducible Representations of Real Algebraic Groups. AMS, Providence (1973) MATH
71.
Langlands, R.P.: On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Math., vol. 544. Springer, Berlin (1976) MATH
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 CrossRef
73.
Langlands, R.P.: Les débuts d’une formule des traces stables. Publ. math. de l’Université Paris, vol. VII (1983) MATH
74.
75.
Laumon, G.: Chtoucas de Drinfeld et correspondance de Langlands. Gaz. Math. 88, 11–33 (2001) MATH
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
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
78.
Laumon, G., Rapoport, M., Stuhler, U.: Elliptic sheaves and the Langlands correspondence. Invent. Math. 113, 217–338 (1993) MathSciNetCrossRefMATH
79.
Lusztig, G.: Some examples of square-integrable representations of semisimple \(p\)-adic groups. Trans. Am. Math. Soc. 277, 623–653 (1983) MathSciNetMATH
80.
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) CrossRefMATH
82.
Mœglin, C., Waldspurger, J.-L.: Stabilisation de la formule des traces tordue. Progress in Math., vol. 316–317. Birkhäuser, Boston (2017) MATH
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
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)
86.
87.
Ngô, B.C.: Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111, 1–169 (2010) CrossRefMATH
88.
Renard, D.: Représentations des groupes réductifs \(p\)-adiques, Cours spécialisés. Société Mathématique de France, vol. 17 (2010) MATH
89.
Scholze, P.: The local Langlands correspondence for \(\mathrm{GL} _{n}\) over \(p\)-adic fields. Invent. Math. 192, 663–715 (2013) MathSciNetCrossRefMATH
90.
Scholze, P.: Perfectoid spaces: a survey. In: Current Developments in Mathematics 2012, pp. 193–227. International Press, Somerville (2013)
91.
Serre, J.-P.: Cours d’arithmétique. P.U.F, Paris (1970) MATH
92.
Springer, T.A.: Linear Algebraic Groups, 2nd edn. Progress in Math., vol. 9. Birkhäuser, Boston (1998) CrossRefMATH
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)
94.
Vignéras, M.-F.: Représentations \(\ell \)-modulaires d’un groupe réductif \(p\)-adique avec \(\ell \ne p\). Progr. Math., vol. 137. Birkhäuser, Boston (1996)
95.
Vignéras, M.-F.: Correspondance de Langlands semi-simple pour \(\mathrm{GL} (n,F)\) modulo \(\ell \ne p\). Invent. Math. 144, 177–223 (2001) MathSciNetCrossRefMATH
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) CrossRef
97.
Waldspurger, J.-L.: Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondammental. Can. J. Math. 43, 852–896 (1991) CrossRefMATH
98.
Waldspurger, J.-L.: L’endoscopie tordue n’est pas si tordue. Mem. Amer. Math. Soc., vol. 194 (2008) MATH
99.
Weil, A.: Adeles and Algebraic Groups. Progress in Math., vol. 23. Birkhäuser, Basel (1982) CrossRefMATH
Metadaten
Titel
Around the Langlands Program
verfasst von
Anne-Marie Aubert
Publikationsdatum
13.09.2017
Verlag
Springer Berlin Heidelberg
Erschienen in
Jahresbericht der Deutschen Mathematiker-Vereinigung / Ausgabe 1/2018
Print ISSN: 0012-0456
Elektronische ISSN: 1869-7135
DOI
https://doi.org/10.1365/s13291-017-0171-8