Content 

 

Abelian varieties and Shimura varieties provide rich sources for current research in Arithmetic Geometry. Main parts of this course

will focus on the followings: basics on Shimura varieties, introductory on the canonical models, review of Class Field Theory,

complex multiplication, some techniques in Delignes construction,isocrystals with additional structures, and several constructions

of Kottwitz for studying isocrystals with additional structures.

 

We also hope to introduce notion of Tannakian categories and Hodge theory, especially on the theory of absolute Hodge cycles, or likely put them in a future course. Part of this course will discuss ongoing research projects including mass formulas for definite unimodular Hermitian forms, and discuss some papers on local densities.

 

References

 

[1] Deligne’s 2 papers on Shimura varieties (Sem. Bourbaki talk and Corvallis)

[2] Kottwitz’s papers:

 

       (a) Isocrystals with additional structure, Compos. Math. 56 (1985), 201-220.

       (b) Isocrystals with additional structure II, Compos. Math. 109 (1997), 255-339

       (c) Points on some Shimura varieties over finite fields. JAMS. 5 (1992) 373-444.

 

[3] Hashimoto and Koseki, Unimodular definite Hermitian Forms I. Tohoku 41 (1989), 1-30. 

 

Ps. The classes of 5/9, 5/12, 5/16, 5/19, 5/30 will be rescheduled to 4/19, 4/26, 5/3, 5/24, 5/31 (Wed) 2-5pm with each 3-hour (3 50 mins classes).