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 O-modules and Lubin-Tate deformation spaces.
  • [scan] 17 May (James), p-adic 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), p-divisible groups with complex multiplication.
  • [scan] 14 June (Wansu), Rapoport-Zink spaces of Scholze-Weinstein.

    [Lent 2013]

  • [scan] 25 Jan (Wansu), Faltings deformation spaces and the Rapoport-Zink 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), Ihara-Langlands-Kottwitz 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 Weil-Deligne 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 (Jean-Francois Dat), What is mod l local Langlands correpondence? (Number Theory Seminar)
  • [scan] 23 Nov (Jean-Francois Dat), The l-adic/mod l local Langlands correpondence and Lubin-Tate 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), Weight-monodromy 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 p-divisible groups (viewed through Vasiu-Zink windows over various frames).
  • [scan] 10 Feb (Wansu), continued.
  • [scan] 17 Feb (Wansu), continued.
  • [scan] 24 Feb (Teruyoshi), Barsotti-Tate (p-divisible) groups, Rapoport-Zink spaces [the first half].
  • [scan] 2 Mar (James), Drinfeld upper half spaces.
  • [scan] 9 Mar (Teruyoshi), Affine Hecke algebras for GL_n and Lubin-Tate spaces.
  • [scan] 16 Mar (James), Drinfeld upper half spaces II.

    [Michaelmas 2011]

  • [scan] 14 Oct (Teruyoshi), Why semistable schemes; l-adic etale cohomology.
  • [scan] 21 Oct (James), Vanishing cycles, semistable curves and Picard-Lefschetz.
  • [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 Barsotti-Tate groups.
  • [scan] 18/25 Nov (Wansu), continued.
  • [scan] 18 Nov (Teruyoshi), Weight spectral sequences, action of correspondences.
  • [scan], [Tony's] 15 Nov (Tony), p-adic comparison theorems and Beilinson's new proof.
  • [scan] 2 Dec (Tony), continued.

    ______references

    [Easter 2013]

    Wansu:

    Coordinate modules:

  • [pdf] P. Gabriel, SGA3-I, Expos'e VIIA: Etude infinit'esimale des sch'emas en groupes, 1970.

    Rapoport-Zink spaces by Scholze-Weinstein:

  • [pdf] P. Scholze, J. Weinstein, Moduli of p-divisible groups, 2012.
  • [pdf] T. Wedhorn, Adic spaces, 2012.

    James:

    p-adic Hodge theory by Scholze:

  • [pdf] P. Scholze, p-adic Hodge theory for rigid analytic varieties, 2012.
  • [pdf] P. Scholze, Perfectoid spaces: a survey, 2013.

    Chris:

    p-divisible 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:

    Ihara-Langlands-Kottwitz method:

  • [pdf] R. Langlands, Modular forms and l-adic 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 L-functions, Vol. II, 1990.
  • [pdf] R. Kottwitz, lecture notes.
  • [pdf] P. Scholze, The Langlands-Kottwitz approach for the modular curve, 2010.
  • [pdf] P. Scholze, The Langlands-Kottwitz 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 L-functions, Part 2, 1979.
  • [pdf] R. Langlands, Automorphic representations, Shimura varieties, and motives. Ein M"archen, in: Automorphic Forms, Representations, and L-functions, Part 2, 1979.
  • [pdf] L. Clozel, Motifs et formes automorphs, in: Automorphic Forms, Shimura Varieties, and L-functions, 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 L-functions, Part 2, 1979.

    De Rham cohomology and Gauss-Manin 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 L-functions, 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 p-divisible 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 Langlands-Rapoport 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, F-isocrystals and de Rham cohomology I, 1983.
  • [pdf] A. Ogus, F-isocrystals and de Rham cohomology. II. Convergent isocrystals, 1984
  • [pdf] G. Prasad, J.K. Yu, On quasi-reductive group schemes, 2006.
  • [pdf] A. Ogus, A p-adic analogue of the Chowla-Selberg formula, in: p-adic Analysis, 1990.
  • [pdf] D. Blasius, A p-adic 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 non-abelian Lubin-Tate 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 p-divisible group, 2002.
  • [pdf] A. Grothendieck, Groupes de Barsotti-Tate 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:

    p-divisible 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 Vasiu-Zink 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 p-divisible groups over regular local rings of arbitrary dimension, 2010.

  • [pdf] T. Zink, A Dieudonn'e theory for p-divisible groups, 2001.
  • [pdf] T. Zink, Windows for displays of p-divisible groups, 2001.
  • [pdf] T. Zink, The display of a formal p-divisible group, 2002.
  • [pdf] C.-L. Chai, Notes on Cartier-Dieudonn'e theory, 2004.

  • [pdf] L. Illusie, Deformations de Groupes de Barsotti-Tate, 1985.
  • [pdf] J. de Jong, Barsotti-Tate 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, p-adic uniformisation of Shimura curves: the theorems of Cherednik and Drinfeld (translation), 1991.
  • [pdf] Y. Mieda, Introduction to p-adic uniformisation of Shimura curves, notes.

    [Michaelmas 2011]

    Teruyoshi:

    pure motives:

  • [pdf] U. Jannsen, Motives, numerical equivalence, and semi-simplicity, 1992.

    6 operations, l-adic 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 p-adiques, 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, Epsilon-factor 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 l-adique de l'accouplement de monodromie (d'apr`es Grothendieck), in: Courbes modulaires et courbes de Shimura, 1991.
  • [pdf] L. Illusie, Sur la formule de Picard-Lefschetz, in: Algebraic Geometry 2000, Azumino, 2002.
  • [pdf] P. Deligne, SGA7-1, Expos'e I, 1972.
  • [pdf] P. Deligne, SGA7-2, Expos'e XIII, 1973.
  • [pdf] P. Deligne, SGA7-2, 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 l-adiques associ'ees aux formes modulaires de Hilbert, 1986.

    Wansu:

    log geometry:

  • [pdf] K. Kato, Logarithmic structures of Fontaine-Illusie, 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, K-theory of log schemes I, 2008.
  • [pdf] A. Ogus, Lectures on logarithmic algebraic geometry, 2006.

    log p-divisible 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 p-divisible groups (d'apr'es Kato), manuscript.

    Tony:

    p-adic Hodge theory:

  • [pdf] J.-M. Fontaine, Le corps des p'eriods p-adiques, in: P'eriodes p-adiques, 1994.
  • [pdf] J.-M. Fontaine, Repr'esentations p-adiques semi-stables, in: P'eriodes p-adiques, 1994.
  • [pdf] O. Hyodo, K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, in: P'eriodes p-adiques, 1994.
  • [pdf] K. Kato, Semi-stable reduction and p-adic etale cohomology, in: P'eriodes p-adiques, 1994.
  • [pdf] O. Brinon, B. Conrad, CMI Summer School notes on p-adic Hodge theory, 2009.

    crystalline cohomology:

  • [pdf] A. Grothendieck, Groupes de Barsotti-Tate 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, F-isocrystals and de Rham cohomology, I, 1983.

    Beilinson's work:

  • [pdf] A. Beilinson, p-adic periods and derived de Rham cohomology, preprint.
  • [pdf] A. Beilinson, On the crystalline period map, preprint.


  • Last modified: 14 June, 2013.