The fundamental problem studied in arithmetic algebraic geometry is the solution of systems of algebraic equations. The notion of an Algebraic Scheme is the essential geometric notion that incorporates this question. We then introduce the notion of vector group schemes and the K-group of such objects. With some additional constraints these are the groups that seem to arise in many cryptographic contexts.
While we cannot hope to introduce all the algebraic geometry and commutative algebra that is necessary to study these K-groups here, we give the fundamental definitions and some important examples. We will also not give proofs as the subject is too vast to be covered here. When we apply this theory to hyper-elliptic curves in the next section we will be more precise.