These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ c
ase) that is based on a quaternion trick. In fact, a a slightly stronger version of those theorems with finite coefficients is proven.
We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.
We prove that an analogue of Jordans theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of Vladimir L. Popov.
