Contents:

This is the second semester of a continued one-year NCTS course in 2015-16 for investigating abelian varieties and related topics. We plan to discuss the following topics: Endomorphism algebras of abelian varieties (of type D), preliminaries on Shimura varieties, preliminaries on Aralekov Geometry, Hilbert modular surfaces and Hilbert modular groups, arithmetic genera and elliptic fixed points, description of certain polarized superspecial abelian varieties over Fp, derived categories and monodromy. Lectures are given in English.  

References:

[1] Endomorphism algberas:

Oort (Endomorphism Algebra of AV, 1987)  

Shimura (Analytic families, Ann. Math. 1963).

[2] Hilbert modular surfaces:

(a) Hirzebruch: Enseignement Math 19(1973).

(b) van der Geer: Hilbert modular surfaces. Springer-Verlag (1988).

(c) Bruinier, Hilbert modular forms and applications. The 1-2-3 modular forms (2008).  

[3] Faltings and Wueztholz: Rational Points (1984).

[4] Papers: Deligne’s 2 papers on Shimura varieties (Sem. Bourbaki talk and Corvallis), Kottwitz (Corvallis, 1979), Cartier (Corvallis, 1979)