study group on
Arithmetic Geometry
This is a website for the study group on Arithmetic Geomtery; here are some material for this study group.
In 2011 Tony and I thought we'd like to have some "study group
at the research level"  and we planned a loosely connected series
of talks & discussions. Some of the talks should be accessible to graduate students. My hope is to have a forum where the speakers can talk
freely on their research questions and have informal discussions,
and at the same time students can get something out of it
by asking questions and watching the speakers discuss.
Meetings: Fri. 2:30  4:30pm, MR 11; followed by cookies in Pav E lounge
First talk (Easter 2013): Fri., 10th May., 2013.
Last talk (Easter 2013): Fri., 14th June, 2013.
______notes
[Easter 2013]
[scan] 10 May (Wansu), Coordinate modules for (Drinfel'd) formal Omodules and LubinTate deformation spaces.
[scan] 17 May (James), padic Hodge theory for rigid analytic spaces by Scholze.
[scan] 24 May (James), continued.
31 Mar (Tony), Local purity for sheaves with geometrically semisimple monodromy implies WMC.
[scan] 7 June (Chris Blake), pdivisible groups with complex multiplication.
[scan] 14 June (Wansu), RapoportZink spaces of ScholzeWeinstein.
[Lent 2013]
[scan] 25 Jan (Wansu), Faltings deformation spaces and the RapoportZink spaces of Hodge type.
[scan] 1 Feb (Wansu), continued.
[scan] 8 Feb (James), Semistable models of modular curves and mod p Galois representations.
[scan] 15 Feb (Nicolas), IharaLanglandsKottwitz method on Shimura varieties (good reduction).
[scan] 22 Feb (Nicolas), continued (bad reduction by Scholze).
[scan] 1 Mar (Wansu), Integral models of Shimura varieties of Hodge type.
[scan] 8 Mar (Wansu), continued.
[scan] 15 Mar (Tony), Mixed WeilDeligne representations and sheaves.
[Michaelmas 2012]
[James' notes] 12 Oct (Teruyoshi), The global Langlands correspondence for GL(n) (overview).
[scan] 19 Oct (Tony), Absolute Hodge cycles.
[scan] 26 Oct (Wansu), Shimura varieties of Hodge type and Hodge tensors.
[scan] 2 Nov (Wansu), Integral models of Shimura varieties of Hodge type.
[scan] 9 Nov (Nicolas), Kisin's key lemma.
[scan] 16 Nov (Nicolas), Faltings' deformation theory.
[scan] 20 Nov (JeanFrancois Dat), What is mod l local Langlands correpondence? (Number Theory Seminar)
[scan] 23 Nov (JeanFrancois Dat), The ladic/mod l local Langlands correpondence and LubinTate spaces.
[scan] 30 Nov (Teruyoshi), Semistable models of modular curves.
[scan] 30 Nov (James), Mod p etale cohomology of modular curves and its semistable model.
7 Dec (Teruyoshi/James), continued.
[Easter 2012]
[notes] 27 Apr (Teruyoshi), Basics on Witt rings
[scan] 4 May (Wansu), Perfectoid spaces (overview).
[scan] 11 May (Wansu), Almost ring theory.
[scan] 18 May (Yoichi Mieda), Adic spaces.
[scan] 25 May (Teruyoshi), Perfectoid affinoids.
[scan] 1 June (James), Etale topology on perfectoid spaces.
[scan] 8 June (James), continued.
[scan] 15 June (James), continued.
[scan] 15 June (Tony), Weightmonodromy conjecture.
[scan] 22 June (Tony), continued.
[Lent 2012]
[scan] 27 Jan (Tony), Beilinson's proof continued (to be continued).
[scan] 3 Feb (Wansu), Dieudonn'e displays for pdivisible groups (viewed through VasiuZink windows over various frames).
[scan] 10 Feb (Wansu), continued.
[scan] 17 Feb (Wansu), continued.
[scan] 24 Feb (Teruyoshi), BarsottiTate (pdivisible) groups, RapoportZink spaces [the first half].
[scan] 2 Mar (James), Drinfeld upper half spaces.
[scan] 9 Mar (Teruyoshi), Affine Hecke algebras for GL_n and LubinTate spaces.
[scan] 16 Mar (James), Drinfeld upper half spaces II.
[Michaelmas 2011]
[scan] 14 Oct (Teruyoshi), Why semistable schemes; ladic etale cohomology.
[scan] 21 Oct (James), Vanishing cycles, semistable curves and PicardLefschetz.
[scan] 28 Oct (Teruyoshi), etale sheaves in derived categories.
[scan] 28 Oct (Teruyoshi), nearby cycles for semistable schemes. (incomplete)
[scan] 4 Nov (James), Galois reps attached to modular forms, semistable modular curves.
[scan] 11 Nov (Wansu), log geometry and log finite flat group schemes / log BarsottiTate groups.
[scan] 18/25 Nov (Wansu), continued.
[scan] 18 Nov (Teruyoshi), Weight spectral sequences, action of correspondences.
[scan], [Tony's] 15 Nov (Tony), padic comparison theorems and Beilinson's new proof.
[scan] 2 Dec (Tony), continued.
______references
[Easter 2013]
Wansu:
Coordinate modules:
[pdf] P. Gabriel, SGA3I, Expos'e VIIA: Etude infinit'esimale des sch'emas en groupes, 1970.
RapoportZink spaces by ScholzeWeinstein:
[pdf] P. Scholze, J. Weinstein, Moduli of pdivisible groups, 2012.
[pdf] T. Wedhorn, Adic spaces, 2012.
James:
padic Hodge theory by Scholze:
[pdf] P. Scholze, padic Hodge theory for rigid analytic varieties, 2012.
[pdf] P. Scholze, Perfectoid spaces: a survey, 2013.
Chris:
pdivisible groups with CM:
[pdf] B. Conrad, Lifting global representations with local properties, 2011.
[Lent 2013]
Wansu:
Artin's approximation:
[pdf] M. Artin, Algebraic approximation of structures over complete local rings, 1969.
[pdf] M. Artin, Versal deformations and algebraic stacks, 1974.
Nicolas:
IharaLanglandsKottwitz method:
[pdf] R. Langlands, Modular forms and ladic representations, in: Modular Functions of One Variable II, 1972.
[pdf] M. Rapoport, On the bad reduction of Shimura varieties, in: Automorphic Forms, Shimura Varieties, and Lfunctions, Vol. II, 1990.
[pdf] R. Kottwitz, lecture notes.
[pdf] P. Scholze, The LanglandsKottwitz approach for the modular curve, 2010.
[pdf] P. Scholze, The LanglandsKottwitz approach for some simple Shimura varieties, 2010.
[Michaelmas 2012]
Teruyoshi:
Global Langlands:
[pdf] R. Taylor, Galois representations, 2004.
[pdf] J.M. Fontaine, B. Mazur, Geometric Galois representations, 1997.
[pdf] J. Tate, Number theoretic background, in: Automorphic Forms, Representations, and Lfunctions, Part 2, 1979.
[pdf] R. Langlands, Automorphic representations, Shimura varieties, and motives. Ein M"archen, in: Automorphic Forms, Representations, and Lfunctions, Part 2, 1979.
[pdf] L. Clozel, Motifs et formes automorphs, in: Automorphic Forms, Shimura Varieties, and Lfunctions, Vol. I, 1990.
[pdf] K. Buzzard, T. Gee, The conjectural connections between automorphic representations and Galois representations, 2011.
Tony (+ new study group):
Motives and absolute Hodge cycles:
[pdf] P. Deligne, J.S. Milne, A. Ogus, K.y. Shih, Hodge Cycles, Motives and Shimura Varieties, 1982.
[pdf] P. Deligne, Hodge cycles on abelian varieties (Intro + Chap. I of the above book).
[pdf] P. Deligne, Valeurs de fonctions L et p'eriodes d'int'egrals, in: Automorphic Forms, Representations, and Lfunctions, Part 2, 1979.
De Rham cohomology and GaussManin connections:
[pdf] N. Katz, T. Oda, On the differentiation of De Rham cohomology classes with respect to parameters, 1968.
[pdf] P. Deligne, Travaux de Griffiths, 1970.
[pdf] A. Grothendieck, On the de Rham cohomology of algebraic varieties, 1965.
Wansu/Nicolas:
Shimura varieties:
[pdf] P. Deligne, Travaux de Shimura, 1971.
[pdf] P. Deligne, Vari'et'es de Shimura, in: Automorphic Forms, Representations, and Lfunctions, Part 2, 1979.
Integral models of Shimura varieties:
[pdf] M. Kisin, Integral canonical models of Shimura varieties, 2009.
[pdf] M. Kisin, Integral models for Shimura varieties of abelian type, 2010.
[pdf] G. Faltings, Integral crystalline cohomology over very ramified valuation rings, 1999.
[pdf, errata] J.S. Milne, The points on a Shimura variety modulo a prime of good reduction, in: The Zeta Function of Picard Modular Surfaces, 1992.
[pdf] B. Moonen, Models of Shimura varieties in mixed characteristics, 1996.
[pdf] A. Vasiu, Th. Zink, Purity reesults for pdivisible groups and abelian schemes over regular bases of mixed characteristic, 2010.
[pdf] K. Madapusi Pera, Toroidal compactifications of integral canonical models of Shimura varieties of Hodge type.
[pdf] A. Vasiu, On the Tate and LanglandsRapoport conjectures for special fibres of integral canonical models of Shimura varieties of abelian type, 2012.
[pdf] M. Kisin, Mod p points on Shimura varieties (slides), 2011.
[pdf] N. Katz, Travaux de Dwork, 1972.
[pdf] P. Berthelot, A. Ogus, Fisocrystals and de Rham cohomology I, 1983.
[pdf] A. Ogus, Fisocrystals and de Rham cohomology. II. Convergent isocrystals, 1984
[pdf] G. Prasad, J.K. Yu, On quasireductive group schemes, 2006.
[pdf] A. Ogus, A padic analogue of the ChowlaSelberg formula, in: padic Analysis, 1990.
[pdf] D. Blasius, A padic property of Hodge classes on abelian varieties, in: Motives, Part 2, 1994.
Teruyoshi/James:
Integral models of modular curves and mod p local Langlands:
[pdf] V.G. Drinfeld, Elliptic Modules I, II, 1974, 1977.
[pdf] B. Edixhoven, Modular parametrizations at primes of bad reduction, preprint, 2001.
[pdf] T. Yoshida, On nonabelian LubinTate theory via vanishing cycles, 2010.
[pdf] B. Edixhoven, Minimal resolution and stable reduction of X_0(N), 1990.
[pdf] B. Edixhoven, The weight in Serre's conjectures on modular forms, 1992.
[pdf] B. Haastert, J.C. Jantzen, Filtrations of the discrete series of SL_2(q) via crystalline cohomology, 1990.
[Easter 2012]
Teruyoshi:
Witt rings:
[pdf] N. Bourbaki, Alg`ebre commutative chapitre IX: Anneau locaux noeth'eriens complets, 1983.
[pdf] T. Zink, The display of a formal pdivisible group, 2002.
[pdf] A. Grothendieck, Groupes de BarsottiTate et cristaux de Dieudonn'e, 1974.
Wansu:
Perfectoid spaces:
[pdf] P. Scholze, Perfectoid spaces, 2011.
[pdf] O. Gabber, L. Ramero, Almost ring theory, 2002.
[pdf] T. Wedhorn, Adic spaces, 2012.
James/Tony:
[pdf] R. Huber, A finiteness result for direct image sheaves on the 'etale site of rigid analytic varieties (parts), 1998.
[Lent 2012]
Wansu:
pdivisible groups and displays:
[pdf] W. Messing, Travaux de Zink, 2006.
[pdf] W. Messing, Slides for Travaux de Zink.
see also Messing's lecture videos from Clay Summer School 2006.
[pdf] E. Lau, A note on VasiuZink windows, preprint.
[pdf] E. Lau, Relations between Dieudonn'e displays and crystalline Dieudonn'e theory, preprint.
[pdf] E. Lau, A duality theorem for Dieudonn'e displays, 2009.
[pdf] E. Lau, Frames and finite group schemes over complete regular local rings, 2010.
[pdf] A. Vasiu, T. Zink, Breuil's classification of pdivisible groups over regular local rings of arbitrary dimension, 2010.
[pdf] T. Zink, A Dieudonn'e theory for pdivisible groups, 2001.
[pdf] T. Zink, Windows for displays of pdivisible groups, 2001.
[pdf] T. Zink, The display of a formal pdivisible group, 2002.
[pdf] C.L. Chai, Notes on CartierDieudonn'e theory, 2004.
[pdf] L. Illusie, Deformations de Groupes de BarsottiTate, 1985.
[pdf] J. de Jong, BarsottiTate groups and crystals, ICM talk, 1998.
[pdf] J. de Jong, Crystalline Dieudonn'e module theory via formal and rigid geometry, 1995.
[pdf, erratum] J. de Jong, Crystalline Dieudonn'e module theory via formal and rigid geometry, 1995.
[pdf] N. Bourbaki, Alg`ebre commutative chapitre IX: Anneau locaux noeth'eriens complets, 1983.
James:
Drinfeld upper half spaces:
[pdf] J.F. Boutot, H. Carayol, padic uniformisation of Shimura curves: the theorems of Cherednik and Drinfeld (translation), 1991.
[pdf] Y. Mieda, Introduction to padic uniformisation of Shimura curves, notes.
[Michaelmas 2011]
Teruyoshi:
pure motives:
[pdf] U. Jannsen, Motives, numerical equivalence, and semisimplicity, 1992.
6 operations, ladic sheaves:
[pdf] A. Grothendieck, SGA5, Expos'e I, 1977.
[pdf] J.P. Jouanolou, SGA5, Expos'e V, 1977.
[pdf] U. Jannsen, Continuous 'etale cohomology, 1988.
absolute purity:
[pdf] K. Fujiwara, A proof of the absolute purity conjecture (after Gabber), in: Algebraic Geometry 2000, Azumino, 2002.
[pdf] L. Illusie, Perversit'e et variation, 2003.
weight spectral sequences:
[pdf] L. Illusie, Autour du th'eor`eme de monodromie locale, in: P'eriodes padiques, 1994.
[pdf] L. Illusie, On semistable reduction and the calculation of nearby cycles, in: Geometric aspects of Dwork theory II, 2004.
[pdf] T. Saito, Weight spectral sequences and independence of l, 2003.
[pdf] T. Saito, Epsilonfactor of a tamely ramified sheaf on a variety, 1993.
[pdf] M. Rapoport, Th. Zink, "Uber die lokale Zetafunktion von Shimuravariet"aten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charkteristik. 1982.
[pdf] T. Yoshida, On the action of algebraic correspondences on weight spectral sequences, preprint.
James:
etale cohomology:
[pdf] E. Freitag, R. Kiehl, Etale cohomology and the Weil conjecture, 1988.
vanishing cycles of curves:
[pdf] L. Illusie, R'ealisation ladique de l'accouplement de monodromie (d'apr`es Grothendieck), in: Courbes modulaires et courbes de Shimura, 1991.
[pdf] L. Illusie, Sur la formule de PicardLefschetz, in: Algebraic Geometry 2000, Azumino, 2002.
[pdf] P. Deligne, SGA71, Expos'e I, 1972.
[pdf] P. Deligne, SGA72, Expos'e XIII, 1973.
[pdf] P. Deligne, SGA72, Expos'e XV, 1973.
modular/Shimura curves:
[pdf] K. Ribet, On modular representations of Gal(Qbar/Q), 1990.
[pdf] H. Carayol, Sur les repr'esentations ladiques associ'ees aux formes modulaires de Hilbert, 1986.
Wansu:
log geometry:
[pdf] K. Kato, Logarithmic structures of FontaineIllusie, 1989.
[pdf] L. Illusie, Logarithmic spaces, 1994.
[pdf] K. Kato, Toric singularities, 1994.
[pdf] K. Kato, C. Nakayama, Log Betti cohomology, Log 'etale cohomology, Log de Rham cohomology of log schemes over C, 1999.
[pdf] C. Nakayama, Nearby cycles for log smooth families, 1998.
[pdf] L. Illusie, An overview of the work of K. Fujiwara, K. Kato and C. Nakayama on logarithmic 'etale cohomology, 2002.
[pdf] W. Niziol, Ktheory of log schemes I, 2008.
[pdf] A. Ogus, Lectures on logarithmic algebraic geometry, 2006.
log pdivisible groups, log abelian varieties:
[pdf] K. Kato, Logarithmic Dieudonne theory, manuscript.
[pdf] K. Kato, Logarithmic degeneration and Dieudonne theory, manuscript.
[pdf] T. Kajiwara, K. Kato, C. Nakayama, Logarithmic abelian varieties, Part I, 2008.
[pdf] T. Kajiwara, K. Kato, C. Nakayama, Logarithmic abelian varieties, Part II, 2008.
[pdf] K.M. Sampath, Log pdivisible groups (d'apr'es Kato), manuscript.
Tony:
padic Hodge theory:
[pdf] J.M. Fontaine, Le corps des p'eriods padiques, in: P'eriodes padiques, 1994.
[pdf] J.M. Fontaine, Repr'esentations padiques semistables, in: P'eriodes padiques, 1994.
[pdf] O. Hyodo, K. Kato, Semistable reduction and crystalline cohomology with logarithmic poles, in: P'eriodes padiques, 1994.
[pdf] K. Kato, Semistable reduction and padic etale cohomology, in: P'eriodes padiques, 1994.
[pdf] O. Brinon, B. Conrad, CMI Summer School notes on padic Hodge theory, 2009.
crystalline cohomology:
[pdf] A. Grothendieck, Groupes de BarsottiTate et cristaux de Dieudonn'e, 1974.
[pdf] P. Berthelot, A. Ogus, Notes on crystalline cohomology, 1978.
P. Berthelot, L. Breen, W. Messing, Th'eorie de Dieudonn'e Cristalline II, 1982.
[pdf] P. Berthelot, A. Ogus, Fisocrystals and de Rham cohomology, I, 1983.
Beilinson's work:
[pdf] A. Beilinson, padic periods and derived de Rham cohomology, preprint.
[pdf] A. Beilinson, On the crystalline period map, preprint.
