UC Berkeley

The thesis is divided into three parts. We consider the essential dimension of algebraic stacks, and compute the essential dimension of moduli stack of polarized K3 surfaces in part 1. In part 2 we concentrate on the period index problems. More precisely, we show that if C is an algebraic curve of genus 1 over a field k of characteristic 0 then the index of C, defined to be the greatest common divisor of the degrees of its closed points, is equal to the index of the Brauer class defined by the G_m-gerbe given by the Picard stack of degree 0 line bundles on C. We also relate this number to the essential dimension. In the last part we give a new proof of the finiteness of abelian varieties over finite fields using the Tate conjecture. This result was first proved by Zarhin.