Seminar on the Proof of Local Langlands

This year, Yihang Zhu and I are organizing a seminar on the work of Harris and Henniart on the proof of the Local Langlands conjectures for GL_n. During this fall semester, the focus will be mainly on the automorphic side of the story. We will save most, if not all, of the geometry for next semester.

Time and place: Tuesdays 4:30-6:00 in Mathematics 507.

Michael Zhao has been Live-TeXing notes for the seminar. They are available on his webpage.


Main References:
[Har] Harris, M. The Local Langlands Conjecture for GL(n) over a p-adic Field, n<p. Invent. Math., 1998.
[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.

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.
[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.

The following three references directly prove the Local Langlands Conjectures, and local-global compatibility at bad primes, by studying certain simple Shimura varieties at those primes. We will not use these references during the first semester, but it is important to know about them.

[HT] Harris, M. and Taylor, R. The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, 2001.
[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.

Tentative Schedule:

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

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]
December 4 Cancelled.
December 11