4th year graduate student at Columbia University. My advisor is Eric Urban.
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.
[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.
[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.
|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||[Har98], [Clo90]|
|October 30||Global Ingredients (continued)||Sam Mundy||[Har98], [Clo90]|
|November 6||No Talk: Election Day|
|November 20||Harris's 1998 Paper||Yihang Zhu||[Har98]|
Right now I am writing a friendly introductory survey of Iwasawa theory, based on lectures I gave at PROMYS in the summer of 2018. They will be available shortly.
Here is my senior thesis (45 pages). The last section contains a new adelic proof of the Riemann-Roch Theorem for number fields.
Here is a link to my paper on this proof on the arXiv (8 pages).
Local Compactness and Number Theory: These are the notes for a seminar course (Math 639 Section 001) which I taught at UNM in the spring of 2015.