Morningside Center of Mathematics
Chinese Academy of Sciences
Morningside Center of Mathematics
Chinese Academy of Sciences
2017 Summer School on Shimura Varieties and Related Topics
September 1122, 2017
Morningside Center of Mathematics, AMSS, CAS, Beijing
Organizers:
Xu Shen (MCM, AMSS, CAS)
ChiaFu 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)
ChiaFu Yu (Academia Sinica)
Chao Zhang (Tsinghua University)
Topics:
Ke Chen: The AndréOort and ColemanOort 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)
ChiaFu Yu: Shimura varieties: canonical models and Newton strata (6 lectures)
Chao Zhang: EO strata and generalized Hasse invariants (3 lectures)
Schedule:

September 11 
September 12 
September 14 
September 15 
10:3011:45 
Yu 
Yu 
Yu 
Zhang 
14:0015:15 
Xue 
Z. Chen 
K. Chen 
Z. Chen 
15:4517:00 
K. Chen 
Xue 
Xue 
K. Chen 

September 18 
September 19 
September 21 
September 22 
10:3011:45 
Yu 
Yu 
Z. Chen 
Yu 
14:0015:15 
Zhang 
Zhang 
Z. Chen 
Xue 
15:4517:00 
Z. Chen 
Xue 
Shen 
Shen 
Abstracts/Outlines:
Ke Chen: The AndréOort and ColemanOort conjectures (3 lectures)
Outline: The AndréOort conjecture describes general behavior of special subvarieties in Shimura varieties. It is a close analogue of the ManinMumford conjecture for tori and abelian varieties, and has seen vast generalizations in the framework of ZilberPink 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 ColemanOort and its interaction with AndréOort.
The three talks are roughly divided as:
(1) introduction to the conjectures of AndréOort and ZilberPink;
(2) approaches in low dimensions;
(3) digression into ColemanOort 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 Fprinciple bundles. Selecta Math. (N.S) 10 (1), 131166 (2004).
2. V. Lafforgue, Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale.
3. Lafforgue, Laurent, Chtoucas de Drinfeld et conjecture de RamanujanPetersson. Astérisque No. 243 (1997).
Xu Shen: Hasse invariants and Galois representations (2 lectures)
Abstract: We will report the works of GoldringKoskivirta 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 EkedahlOort 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 CMmotives.
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) 88:492517.
[4] N. Schappacher, Periods of Hecke characters. Lecture Notes in Mathematics, 1301.
ChiaFu 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), 123165. 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 Lfunctions. Proc. Sympos. Pure Math., 33, Part 2, 247289, 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 semisimple 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 nonvanishing 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 EkedahlOort 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 EkedahlOort stratum a characteristic $p$ modular form on its closure (i.e. Hasse invariant), whose nonvanishing 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. 9671012, 2010.
[3] R. Pink; T. Wedhorn; P. Ziegler: $F$zips with additional structure, Pacific J. Math. 274, no. 1, pp. 183236, 2015.
[4] C. Zhang: EkedahlOort strata for Shimura varieties of Hodge type arXiv:1312.4869v3.
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