| In this dissertation we investigate the structure and moduli spaces of some algebraic K3 surfaces with high rank. In particular, we study K3 surfaces X which have a Shioda-Inose structure, that is, such that X has an involution iota which fixes any regular 2-form, and the quotient X/{lcub}1, iota{rcub} is birational to a Kummer surface.; We can specify the moduli spaces of K3 surfaces with Shioda-Inose structures by identifying them as lattice-polarized K3 surfaces for the lattice E8(-1)2, with the additional data of an ample divisor class. Similarly, we can give the quotient Kummer surface the structure of an E8(-1) ⊕ N-lattice polarized K3 surface, with the additional data of an ample divisor class.; One of the main results is that there is an isomorphism of the moduli spaces of these two types of lattice-polarized K3 surfaces.; When X is an elliptic K3 surface with reducible fibers of types E8 and E7, we describe the Nikulin involution and quotient map explicitly, and identify the quotient K3 surface as a Kummer surface of a Jacobian of a curve of genus 2. Our second main result gives the algebraic identification of the moduli spaces explicitly in this case. |
