On the quasi group of a cubic surface over a finite field

We construct nontrivial homomorphisms from the quasi group of some cubic surfaces over Fp into a group. We show experimentally that the homomorphisms constructed are the only possible ones and that there are no nontrivial homomorphisms in the other cases. Thereby, we follow the classification of cubic surfaces, due to A.Cayley. 1 The quasi group of a cubic surface 1.1. –––– According to Yu. I. Manin [Ma], a cubic surface V carries a structure of a quasi group. For us, this shall simply mean the ternary relation [x1, x2, x3] ⇐⇒ x1, x2, x3 non-singular, intersection of V with a line . If V is defined over a field K then, on V (K), there is a structure of a quasi group. 1.2. –––– Here, the precise definition is that the lines lying entirely on the surface shall not cause any relation. On the other hand, it is allowed that two or all three points coincide. Then, the line shall simply be tangent to the surface of order two or three. 1.3. Definition. –––– Let (Γ, [ ]) be a quasi group and (G,+) be an abelian group. By a homomorphism p : Γ → G, we mean a mapping such that, for a suitable g ∈ G, p(x1) + p(x2) + p(x3) = g whenever [x1, x2, x3] is true. ∗Mathematisches Institut, Universität Bayreuth, Univ’straße 30, D-95440 Bayreuth, Germany, [email protected], Website: http://www.staff.uni-bayreuth.de/∼btm216 ‡Fachbereich 6 Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany, [email protected], Website: http://www.uni-math.gwdg.de/jahnel