Page Title

Seminar on the proof of the local Langlands correspondence for GL(n) over p-adic number fields

In this seminar, we want to understand Scholze's proof of the local Langlands conjecture for GL(n) over p-adic fields, cf. [Sch13], which simplifies substantially some arguments in the proof given by Harris-Taylor, cf. [H-T01]. The proof uses some global arguments, which is based on the study of some unitary Shimura varieties.

Time and Place:

7pm-8:30pm EST, online.

Date	Title	Speaker	Notes
Dec22	Weil-Deligne representations and L-, \epsilon-factors.	Kenta Suzuki	TBA
Dec29	Smooth Reprensetations of GL(n) and the Bernstein Center.	Zeyu Wang	notes
Jan2	Jacquet-Langlands correspondence and Clozel's base change.	Gefei Dang	notes
Jan5	Harris-Taylor's simple Shimura varieties	James Yang	notes
Jan12	Deformation spaces of p-divisible groups and the test functions	Xinyu Zhou	notes
Jan17	Descent properties of the test functions	Kenta Suzuki	TBA
Jan19	Langlands-Kottwitz method	Eunsu Hur	TBA
Jan24	\ell-adic Galois representations attached to automorphic forms	Hao Peng	notes
Jan26	Bijective correspondence for irreducible supercuspidal representations	Danielle Wang	TBA
Jan30	Compatibility of the correspondence	Kenta Suzuki	TBA

Syllabus

References:

Main References:

[A-C89]J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula. Ann. of Math. Stud., 120. Princeton University Press, Princeton, NJ, 1989. xiv+230 pp.
[H-T01]M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Ann. of Math. Stud. 151, Princeton Univ. Press, 2001.
[Kot92]R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373-444.
[Sch13]P. Scholze, The local Langlands correspondence for GL(n) over p-adic fields, Invent. Math. 192 (2013), no.3, 663-715.

Other References:

[Ber84]J. N. Bernstein, Le "centre" de Bernstein, in "Representations of reductive groups over a local field", Travaux en Cours, Hermann, Paris, 1984, 1-32.
[Ber96]V. Berkovich, Vanishing cycles for formal schemes. 2. Invent. Math.125(1996), no.2, 367-390.
[Bum97]D. Bump, Automorphic forms and representations. Cambridge Stud. Adv. Math., 55. Cambridge University Press, Cambridge, 1997. xiv+574 pp.
[B-G14]K. Buzzard and T. Gee, The conjectural connections between automorphic representations and Galois representations. Automorphic forms and Galois representations. Vol. 1, 135-187. London Math. Soc. Lecture Note Ser., 414. Cambridge University Press, Cambridge, 2014.
[B-Z76]J. N. Bernstein and A. V. Zelevinski\v i, Representations of the group GL(n; F), where F is a local non-Archimedean field. Uspehi Mat. Nauk31(1976), no.3, 5-70.
[B-Z77]J. N. Bernstein and A. V. Zelevinski\v i, Induced representations of reductive p-adic groups. 1. Ann. Sci. École Norm. Sup. (4)10(1977), no.4, 441-472.
[Del72]P. Deligne, Les constantes des équations fonctionnelles des fonctions L, in "Modular functions of one variable, 2" (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 349. Springer, Berlin, 1973, 501-597.
[Far]L. Fargues, Motives and automorphic forms: the Abelian case.
[Har93]M. Harris, The local Langlands conjecture for GL(n) over a p-adic field, n< p. Invent. Math.134(1998), no.1, 177-210.
[Hen93]G. Henniart, Caractérisation de la correspondance de Langlands locale par les facteurs \epsilon de paires. Invent. Math.113(1993), no.2, 339-350.
[Hen00]G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math.139(2000), no.2, 439-455.
[JPS83]H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions. Amer. J. Math.105(1983), no.2, 367-464.
[Kud91]S. S. Kudla, The local Langlands correspondence: the non-Archimedean case. Motives (Seattle, WA, 1991), 365-391. Proc. Sympos. Pure Math., 55, Part 2. American Mathematical Society, Providence, RI, 1994.
[Lan]R. P. Langlands, On the functional equation of the Artin L-functions, http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands
[Sch11]P. Scholze, The Langlands-Kottwitz approach for the modular curve. Int. Math. Res. Not. IMRN(2011), no.15, 3368-3425.
[Sch13a]P. Scholze, The Langlands-Kottwitz approach for some simple Shimura varieties, Invent. Math. 192 (2013), no.3, 627-661.
[Tat79]J. T. Tate, Number theoretic background. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 3-26. Proc. Sympos. Pure Math., 33. American Mathematical Society, Providence, RI, 1979.
[Wed08]T. Wedhorn, The local Langlands correspondence for GL(n) over p-adic fields, in "School on Automorphic Forms on GL(n)", volume 21 of ICTP Lect. Notes, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008, 237-320.