TITLE: Elliptic curves over finite fields
QUESTION [6 upvotes]: I have basic questions about elliptic curves over finite fields. 

Where to find general references? Hartshorne for instance restricts to algebraically closed ground fields.
Over an arbitrary field $K$, is the right definition of an elliptic curve a smooth proper
curve of genus 1 with a choice of $K$-rational point? 
What is known about the structure of the group of $K$-rational points when $K$ is finite? In particular, how much does it depend on the curve?
Are there simple examples where you can explicitly see all of the $K$-rational points, $K$ finite?

REPLY [2 votes]: I completely agree with the earlier answers. Just two remarks...

For question 3, If $K$ is finite of cardinality $q$, then $E(K)$ is isomorphic to $\mathbf Z/n\mathbf Z\times\mathbf Z/m\mathbf Z$, where $n$ divides gcd$(q-1,m)$.
Concerning your last question, here is a simple example where you can explicitly 'see' all the $K$-rational points without direct computations; I hope it may interest you. Consider the elliptic curve
$$E:Y^2=X^3+1$$
defined over the finite field $K=\mathbf F_p$, where $p=3n+2$ is a prime number. Then, there is a bijection $\varphi:K\to E(K)-\lbrace O\rbrace$ (where $O$ is the point at infinity) given by
$$\varphi(t)=\left((t^2-1)^{2n+1},t\right).$$