Seminar on the Proof of Local Langlands

During the 2018-2019 academic year, Yihang Zhu and I organized a seminar on the proof of the Local Langlands conjectures for GL_n. We focused on the Scholze's work during the spring semester, especially on his GL_2 paper [S11]. During the fall semester, the focus was mainly on the automorphic side of the story, and we worked towards presenting [Har] and [Hen00].

Time and place: Tuesdays 1:00-2:30 in Mathematics 622.


Main References:
[S11] Scholze, P. The Langlands-Kottwitz Approach for the Modular Curve. IMRN, 2011.
[S13-1] Scholze, P. The Langlands-Kottwitz Approach for Some Simple Shimura Varieties. Invent. Math., 2013.
[S13-2] Scholze, P. The Local Langlands Correspondence for GLn over p-adic Fields. Invent. Math., 2013.

Other References:
[Clo90] Clozel, L. Motifs et formes automorphes: applications du principe de fonctorialité. In: Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77-159.
[Del73] Deligne, P. Les constantes des Équations fonctionnelles des fonctions L. 501-597. Lecture Notes in Math., Vol. 349.
[Har] Harris, M. The Local Langlands Conjecture for GL(n) over a p-adic Field, n<p. Invent. Math., 1998.
[HT] Harris, M. and Taylor, R. The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, 2001.
[Hen93] Henniart, G. Caractérisation de la correspondance de Langlands locale par les ε facteurs de paires. Invent. Math., 1993.
[Hen00] Henniart, G. Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math., 2000.
[JPSS83] Jacquet, H., Piatetskii-Shapiro, I., and Shalika, J. Rankin-Selberg Convolutions. Amer. J. Math., 1983.
[Wed08] Wedhorn, T. The Local Langlands Correspondence for GL(n) over p-adic Fields. Available online, by clicking the link provided.

Tentative Schedule:

Yihang has created a program for the seminar containing instructions for upcoming talks.

Date Title Speaker References
February 5 Henniart's 2000 Paper Yihang Zhu [Hen00]
February 12 Hecke Algebras and Orbital Integrals Robin Zhang [S11] Ch. 2-3
February 19 Geometry of Modular Curves and Counting Points Stan Atanasov [S11] Ch. 4-5
February 26 Counting Points, Continued Stan Atanasov [S11] Ch. 6
l-adic Galois Representations Sam Mundy [S11] Ch. 7
March 5 l-adic Galois Representations, Continued Sam Mundy [S11] Ch. 7
March 12 More on l-adic representations Sam Mundy [S11] Ch. 7
March 19 No Talk: Spring Break
March 26 Nearby Cycles Shizhang Li [S11] Ch 8
April 2 Traces as Twisted Orbital Integrals Yihang Zhu [S11] Ch 9
April 9 Cancelled
April 16 Cancelled
April 23 Scholze's Local Langlands Paper Yihang Zhu [S13-2]
April 30 Scholze's Local Langlands Paper, Continued Yihang Zhu [S13-2]
May 6 One off: Skinner-Urban on the Main Conjecture for GL_2 (Monday at 1:00 in 507) Sam Mundy Abstract
May 13 More on Scholze's Local Langlands Paper (Monday at 1:30 in 507) Yihang Zhu [S13-2]

Last Semester's Schedule:

Date Title Speaker References
September 11 Introductory Talk Michael Harris
September 18 Review: Admissible Representations and Weil-Deligne Representations Stanislav Atanasov [Wed08]
September 25 Review: L- and ε-factors Robin Zhang [Wed08], [Del73]
October 2 Henniart's Characterization Paper Yu-Sheng Lee [Hen93]
October 9 Jacquet--Piatetskii-Shapiro--Shalika Hervé Jacquet [JPSS83]
October 16 Automorphic Forms and Automorphic Representations Sam Mundy Your orals syllabus.
October 23 Global Ingredients Sam Mundy [Clo90], [Har98]
October 30 Global Ingredients (continued) Sam Mundy [Clo90], [Har98]
November 6 No Talk: Election Day
November 13 Cancelled.
November 20 Harris's 1998 Paper Yihang Zhu [Har98]
November 27 Harris's 1998 Paper (continued) Yihang Zhu [Har98]