2017 Summer School on Shimura Varieties and Related Topics

September 11-22, 2017

Morningside Center of Mathematics, AMSS, CAS, Beijing

Organizers: 

    Xu Shen (MCM, AMSS, CAS)

    Chia-Fu Yu (Academia Sinica)

Place:

    Room 818 of South Building of AMSS

Speakers:

Ke Chen (Nanjing University)

Zongbin Chen (Tsinghua University)

Xu Shen (AMSS, Chinese Academy of Sciences)

Jiangwei Xue(Wuhan University)

Chia-Fu Yu (Academia Sinica)

Chao Zhang (Tsinghua University)

Topics:

Ke Chen: The André-Oort and Coleman-Oort conjectures (3 lectures)

Zongbin Chen: Introduction to Langlands correspondence for the function fields (5 lectures)

Xu Shen: Hasse invariants and Galois representations (2 lectures)

Jiangwei Xue: CM theory (5 lectures)

Chia-Fu Yu: Shimura varieties: canonical models and Newton strata (6 lectures)

Chao Zhang: EO strata and generalized Hasse invariants (3 lectures)

Schedule:

Abstracts/Outlines:

Ke Chen: The André-Oort and Coleman-Oort conjectures (3 lectures)

Outline:  The André-Oort conjecture describes general behavior of special subvarieties in Shimura varieties. It is a close analogue of the Manin-Mumford conjecture for tori and abelian varieties, and has seen vast generalizations in the framework of Zilber-Pink conjectures. In this series of talks we present the

formulations of these conjectures, discuss some known approaches through examples in low dimensions. If time permits we mention the problem of Coleman-Oort and its interaction with André-Oort.

The three talks are roughly divided as:

(1) introduction to the conjectures of André-Oort and Zilber-Pink;

(2) approaches in low dimensions;

(3) digression into Coleman-Oort problem.

Zongbin Chen: Introduction to Langlands correspondence for the function fields (5 lectures)

Abstract:  The moduli stack of Chtoucas is an analogue of the Shimura variety over the function fields, they were introduced by Drinfeld to prove Langlands correspondence for GL_2 in the function field case. In this lecture series, we will try to explain the recent work of V. Lafforgue on the Langlands correspondence in the function field case.

References:

1.    Varshavsky,  Moduli spaces of F-principle bundles. Selecta Math. (N.S) 10 (1), 131-166 (2004).

2.    V. Lafforgue,  Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale.

3.    Lafforgue, Laurent,  Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson. Astérisque No. 243 (1997).

Xu Shen: Hasse invariants and Galois representations (2 lectures)

Abstract:  We will report the works of Goldring-Koskivirta and Boxer on the construction of congruences and Galois representations in the coherent cohomology of certain Shimura varieties, using some generalized Hasse invariants adapted to the Ekedahl-Oort stratification in good reductions of these Shimura varieties.

References:

[1] W. Goldring, J.-S. Koskivirta,  Strata Hasse invariants, Hecke algebras and Galois representations,  arXiv:1507.05032

[2] G. Boxer,  Torsion in the coherent cohomology of Shimura varieties and Galois representations,  arXiv:1507.05922

Jiangwei Xue: CM theory (5 lectures)

Outline:  We'll sketch a proof of the fundamental theorem of complex multiplication, which describes how automorphisms of the field of complex numbers C act on the abelian varieties with complex multiplication and their torsion points. The exposition will follow mostly that of J.S. Milne [1], with the additional input from [2], [3], [4]. We start by reviewing the background material such as abelian varieties with complex multiplication and reflex norms, then develop the fundamental theorem over the reflex field, and finally extend it to the rational field Q.  If time permits, we'll explain the Taniyama group and CM-motives.

References: 

[1] J.S. Milne, The fundamental theorem of complex multiplication, 2007, http://www.jmilne.org/math/articles/2007c.pdf

[2] J.S. Milne, Complex multiplication, Preliminary version, 2006, http://www.jmilne.org/math/CourseNotes/cm.html

[3] J.P. Serre and J. Tate,  Good reductions of abelian varieties,  Ann. of Math. (2) 88:492-517.

[4] N. Schappacher, Periods of Hecke characters.  Lecture Notes in Mathematics, 1301.

Chia-Fu Yu: Shimura varieties: canonical models and Newton strata (6 lectures)

Outline:  The goal of these lecture series is discuss the techniques and ideas in the construction of Shimura varieties of abelian type. This is main content of Deligne’s paper [2]. We will start with definition of Shimura varieties and basic background and discuss briefly the Deligne’s paper [1] where Deligne gave the proof of the existence of canonical models for Shimura varieties of Hodge. Then go into the classification of Shimura varieties of abelian type. We like to discuss quaternionic Shimura varieties as examples. Then for the remaining time we plan to focus on the integral models and Newton strata of Shimura varieties of Hodge type.

References:

[1] P. Deligne,  Travaux de Shimura. Seminaire Bourbaki 389 (1970/71), 123--165. LNM 244, 1971.

[2] P. Deligne,  Vari\'et\'es de Shimura: interpr\'etation modulaire, et techniques de construction de mod\`eles canoniques. Automorphic forms, representations and L-functions. Proc. Sympos. Pure Math., 33, Part 2, 247--289, 1979

[3] B. Moonen,  Models of Shimura varieties in mixed characteristics.  Galois representations in arithmetic algebraic geometry (Durham, 1996), 267—350.

[4] H. Reimann,  The semi-simple zeta function of quaternionic Shimura varieties. LNM 1657, 1997.

Chao Zhang: EO strata and generalized Hasse invariants (3 lectures)

Outline:  Historically, the Hasse invariant is a characteristic $p$ modular form on the good reduction of a modular curve, and its non-vanishing locus is precisely the ordinary locus. The goal of these lectures is to explain a great generalization of this fact. 

After a brief review of Kisin's construction of good reductions of Shimura varieties of Hodge type (see [2]), we will explain the theory of Ekedahl-Oort stratifications following [4]. For a modular curve, this stratification is just the decomposition into the union of ordinary locus and supersingular locus. We will also introduce zip data (see [3]) and explain the group theoretic technics developped in [1] section 4,5. Finally, we will explain, following [1] section 6, how to construct for each Ekedahl-Oort stratum a characteristic $p$ modular form on its closure (i.e. Hasse invariant), whose non-vanishing locus is precisely the stratum.

We refer to Xu Shen's talks for interesting arithmetic applications of this theory.  

References:

[1] W. Goldring; J.-S. Koskivirta: Strata Hasse invariants, Hecke algebras and Galois representations, arXiv:1507.05032.

[2] M. Kisin: Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23, pp. 967-1012, 2010.

[3] R. Pink; T. Wedhorn; P. Ziegler: $F$-zips with additional structure, Pacific J. Math. 274, no. 1, pp. 183-236, 2015.

[4] C. Zhang: Ekedahl-Oort strata for Shimura varieties of Hodge type arXiv:1312.4869v3.