• Speaker(s)Beijing International Center for Mathematical Research
• DateFrom 2011-07-11 To 2011-08-26
• Venue理科一号楼 数学系 多功能教室 1556

BICMR Summer School on Arithmetic Geometry

July--August, 2011

Organizers:

• Ruochuan Liu （刘若川）
• Chenyang Xu（许晨阳）
• Zhiwei Yun（恽之玮）

Time:

• Topic I: July 11 - July 29. (Meet every afternoon during weekdays)
  - Lecture for Topic I: 2:00 pm - 3:30pm;
  - Tea Break: 3:30 pm - 4:00 pm;
  - Lecture for Topic II: 4:00 pm - 5:30 pm;
• Topic II: August 2 - August 4
• Topic III: August 8 - August 26

Room: Topic I/III：理科一号楼 数学系 多功能教室 1556
Topic II: 北京国际数学研究中心 1328房间

Topic I. Deligne-Illusie Theory and Crystalline Cohomology

References
[1] P.Deligne and L.Illusie,
Relèvements modulo p^2 et décomposition du complexe de de Rham. Invent. Math. 89 (1987), no. 2, 247-270.
[2] The chapter written by Illusie in the following book:
Introduction to Hodge theory, by J.Bertin, J-P. Demailly, L.Illusie and C. Peters. SMF/AMS Texts and Monographs, 8.
[3] L.Illusie,
Crystalline cohomology, in Motives, Proc. Sympos. Pure Math., 55, Part 1
[4] P.Berthelot and A.Ogus
Notes on crystalline cohomology.
[4] L.Hesselholt and I.Madsen.
The de Rham-Witt complex in mixed characteristic, available at http://www.math.nagoya-u.ac.jp/~larsh/papers/013/.
[5] L.Illusie.
Complexe de de Rham-Witt et cohomologie cristalline. Annales scientifiques de l’Ecole Normale Superieure, 12(4):501-661, 1979.
[4] A.Ogus and V.Vologodsky,
Nonabelian Hodge theory in characteristic p. Publ. Math. Inst. Hautes Études Sci. No. 106 (2007), 1-138.
 

Tentative Schedule

• 7/11 --- Overview
• 7/12 --- Deligne-Illusie’s paper I
• 7/13 --- Deligne-Illusie’s paper II
• 7/14 --- Deligne-Illusie’s paper III
• 7/15 --- Dwork’s proof of rationality of zeta functions I
• 7/18 --- Dwork’s proof of rationality of zeta functions II
• 7/19 --- Crystalline cohomology I
• 7/20 --- Crystalline cohomology II
• 7/21 --- Crystalline cohomology III
• 7/22 --- DeRham-Witt complex
• 7/25 --- Comparison Theorem I
• 7/26 --- Comparison Theorem II
• 7/27 --- Nonabelian Hodge theory mod p I
• 7/28 --- Nonabelian Hodge theory mod p II
• 7/29 --- Nonabelian Hodge theory mod p III

Topic II. Geometry of Shimura Varieties

References
[1] M. Goresky's survey article
Compactifications and cohomology of modular varieties.
In the book Harmonic analysis, the trace formula, and Shimura varieties, 551-582, Clay Math. Proc., 4
[2] W.L.Baily and A.Borel,
Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) 84 (1966) 442-528.
[3] A.Borel and J-P.Serre,
Corners and arithmetic groups. (with appendix by A. Douady and L. Hérault). Comment. Math. Helv. 48 (1973), 436-491.
[4] A.Ash, D.Mumford, M.Rapoport and Y-S.Tai,
Smooth compactifications of locally symmetric varieties. Second edition.
 

Tentative Schedule

• 7/11 --- Overview & Hermitian symmetric spaces
• 7/12 --- Baily-Borel compactification I
• 7/13 --- Baily-Borel compactification II
• 7/14 --- Torus embeddings
• 7/15 --- Combinatorics of Cones
• 7/18 --- Toroidal compactification I
• 7/19 --- Toroidal compactification II
• 7/20 --- Toroidal compactification III
• 7/21 --- Toroidal compactification IV
• 7/22 --- Toroidal compactification V
• 7/25 --- Toroidal compactification VI
• 7/26 --- Examples---Hilbert modular varieties
• 7/27 --- Examples---Siegel modular varieties
• 7/28 --- Examples---Picard modular surfaces
• 7/29 --- Cohomology

Topic III. Berkovich Spaces

References
[1] Michael Temkin's notes: Introduction to Berkovich analytic spapces.
available at http://www.math.huji.ac.il/~temkin/papers/Lecture_Notes_Berkovich_Analytic_Spaces.pdf
[2] Berkovich's
Non-archimedean analytic spaces, available at http://www.wisdom.weizmann.ac.il/~vova/