Title:

Cubic surfaces over finite fields

It is wellknown that the set of rational points on an elliptic curve forms an abelian group. When the curve is given as a plane cubic in Weierstrass form the group operation is defined via tangent and secant operations. Let S be a smooth cubic surface over a field K. Again one can define tangent and secant operations on S. These do not give S(K) a group structure, but one can still ask for the size of a minimal generating set. In Chapter 2 of the thesis I show that if S is a smooth cubic surface over a field K with at least 4 elements, and if S contains a skew pair of lines defined over K, then any nonEckardt Kpoint on either line generates S(K). This strengthens a result of Siksek [20]. In Chapter 3, I show that if S is a smooth cubic surface over a finite field K = Fq with at least 8 elements, and if S contains at least one Kline, then there is some point P > S(K) that generates S(K). In Chapter 4, I consider cubic surfaces S over finite fields K = Fq that contain no Klines. I find a lower bound for the proportion of points generated when starting with a nonEckardt point P > S(K) and show that this lower bound tends to 1/6 as q tends to infinity. In Chapter 5, I define cinvariants of cubic surfaces over a finite field K = Fq with respect to a given Kline contained in S, give several results regarding these cinvariants and relate them to the number of points SS(K)S. In Chapter 6, I consider the problem of enumerating cubic surfaces over a finite field, K = Fq, with a given point, P > S(K), up to an explicit equivalence relation.
