The Langlands correspondencein arithmetic and geometry
Date: August 1 ~ 5, 2016 Place: Rm. 1503, KIAS |
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Speaker: Ana Caraiani (Princeton / Bonn)
Title: Shimura varieties, perfectoid spaces and torsion classes
Abstract: The goal of this mini-course is to describe Shimura varieties from the point of view of p-adic analytic geometry and p-adic Hodge theory. I will focus on the geometry of the Hodge-Tate period morphism, which reduces us to understanding a much simpler geometric object, the associated flag variety. I will start by giving some background on Scholze’s proof that Siegel modular varieties with infinite p-power level are perfectoid and on the construction of the Hodge-Tate period morphism in this case. I will then explain how to extend the Hodge-Tate period morphism to Shimura varieties of Hodge type, how to define a Newton stratification on the flag variety, and how to compute the fibers of the morphism over individual Newton strata. I will end by explaining how the Hodge-Tate period morphism can be used to understand torsion classes in the cohomology of Shimura varieties or more general locally symmetric spaces. The mini-course will rely primarily on joint work with Peter Scholze.
- References:
1) Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2) 182 (2015), no. 3, 945–1066 http://www.math.uni-bonn.de/people/scholze/Torsion.pdf
2) Caraiani and Scholze,On the generic part of the cohomology of compact unitary Shimura varieties
http://arxiv.org/abs/1511.02418
3) Morel, Construction de representations galoisiennes de torsion, Seminaire Bourbaki 2014-15, no. 1102
http://www.bourbaki.ens.fr/TEXTES/1102.pdf
4) Weinstein, Reciprocity laws and Galois representations: recent breakthroughs. Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 1, 1–39 http://math.bu.edu/people/jsweinst/CEB/BAMS.pdf
Speaker: Laurent Fargues (Jussieu)
Title: Geometrization of the local Langlands correspondence
Abstract: In these talks I will give an introduction to my geometrization conjecture of the local Langlands correspondence. I will focus on the GL_1 case explaining how one can use geometric Langlands techniques to prove the conjecture in this case giving a new proof of local class field theory. I will try to state the conjecture in its full generality at the end.
Speaker: Marie-France Vigneras (Jussieu)
Title: Modular representations of p-adic groups
Abstract: We will give an overview of the modulo p representation theory of finite and p-adic reductive groups, with several examples.
- Basic notions:
1) Bushnell and Henniart: The Local Langlands conjecture for GL(2), Chapters 1, 2,3,
2) Cabanes, Enguehard: Representation theory of finite reductive groups, Chapter 6, sections 6.1, 6.2.
3) Vigneras: The pro-p-Iwahori Hecke algebra of a reductive p-adic group I.
- References:
1) Abe, Henniart, Herzig, Vigneras: A classification of irreducible modulo p representations of p-adic reductive groups.
Speaker: Zhiwei Yun (Stanford)
Title: Intersection numbers and higher derivatives of L-functions for function fields
Abstract: In joint work with Wei Zhang, we prove a higher derivative analogue of the Waldspurger formula and the Gross-Zagier formula in the function field setting under some unramifiedness assumptions. Our formula relates the self-intersection number of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of automorphic L-functions for GL(2). In the first talk, I will give motivation and the statement of the formula; in the second talk, I will explain some geometric ideas in the proof of the formula.
Speaker: Xinyi Yuan (Berkeley)
Title: Introduction to the Colmez conjecture
Abstract: The goal of this series of talks is to present some recent developments of the Colmez conjecture about Faltings heights of abelian varieties with complex multiplication. In the first talk, I will introduce the Colmez conjecture, including its averaged version recently proved by Yuan--Zhang and Andreatta--Goren--Howard--Madapusi-Pera. In the second talk, I will introduce the application of the averaged Colmez conjecture to the Andre--Oort conjecture by Tsimerman. In the third talk, I will sketch the proof of the averaged Colmez conjecture by Yuan--Zhang.