●INTRODUCTION
The theory of almost etale extensions was developed by G. Faltings to prove a series of famous conjectures by Fontaine in p-adic Hodge theory, which predicted deep relations between p-adic etale cohomology and de Rham (or crystalline) cohomology for proper smooth varieties over a p-adic local field. Thanks to the work of many people (Faltings, Fontaine, Hyodo, Kato, Messing, Tsuji, …), these conjectures have become theorems. Among all the known approaches, that of Faltings has the advantage that it can also deal with cases of variant coefficients. In recent years, by pursuing Faltings' idea of almost etale extensions, many advances have been made in understanding p-adic Galois representations in more general geometric settings.The aim of this two-week workshop is to understand the theory of almost etale extensions and its applications to p-adic comparison theorems and p-adic Hodge theory in the relative case. In a series of lectures given by several young researchers, we will explain basic notions in "almost mathematics", Faltings' purity theorem, proof of the comparison theorem between p-adic etale cohomology and crystalline cohomology in the good reduction case, theory of (φ, Γ)-modules in the relative case, and also generalized Tsen theory (if time permits). Prerequisites for these lectures include commutative algebra, algebraic geometry, and basic notions of etale cohomology and crystalline cohomology; in addition, a basic knowledge on classical p-adic Hodge theory (e.g. history of the comparison theorem, classification of p-adic Galois representations of a local field, classical (φ, Γ)-modules, etc.) is crucial to understand the motivations of the problem.
Speakers:
·Miaofen Chen, University Paris-sud 11, France
·Fucheng Tan, Massachusetts Institute of Technology (MIT), USA
·Yichao Tian, Princeton University, USA
·Jilong Tong, Ecole Polytechnique, France
·Shanwen Wang, Ecole Polytechnique, France
·Weizhe Zheng, Columbia University, USA
Organizers:
·Yichao Tian: yichaot@princeton.edu
·Weizhe Zheng: zheng@math.columiba.edu
Schedule:
Schedule.pdf
References:
·F. Andreatta & A. Iovita, Comparison isomorphisms for formal schemes, preprint (2008), available at: http://www.mathstat.concordia.ca/faculty/iovita/research.html
·F. Andreatta & A. Iovita, Global applications of relative (φ, Γ)-modules I, preprint (2006), available at: http://arxiv.org/abs/math/0610692
·G. Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1998), 255-288.
·G. Faltings, Almost etale extensions, Asterisque 279 (2002), 185-270.
·O. Gabber & L. Rameo, Almost ring theory, LNM 1800, Springer, 2002, available at: http://arxiv.org/abs/math/0201175
·M. Olsson, On Faltings' method of almost etale extensions, in Algebraic Geometry: Seattle 2005, AMS, 2009, 811-936, available at: http://math.berkeley.edu/~molsson/
●PAPERS
●REPORTS
On the André-Oort conjecture (2009-8-12)
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On the Albanese morphism (2009-8-12)
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On a prime number theorem for Rankin-Selberg L-functions over different fields (2009-8-11)
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Relative trace formula and arithmetic Gross-Prasad (2009-8-11)
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On the Fourier Coefficients of cuspidal representations of GL(2) (2009-8-10)
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Relative trace formula and Gross-Zagier theorem (2009-8-10)
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