Ternary cubic forms and central simple algebras of degree 3
Fix a ground field F of characteristic neither 2 nor 3 and consider pairs (A,V) consisting of a degree 3 central simple F-algebra A and a 3-dimensional subspace V of the reduced trace zero elements of A which is totally isotropic for the trace quadratic form. Mapping an element of V to its cube defines a cubic form. This thesis is devoted to the classification of such cubic pairs - i.e. the description of a representative of each isomorphism class of cubic pairs - and the study of the associated cubic forms.
First we study some geometrical aspects of ternary cubic forms in general; i.e. we study cubic curves in a projective plane. Apart from the well-known flexes of such a curve, we observe the existence of other special points and lines, which we call Hessian points, harmonic points and harmonic polars, and exhibit their remarkable properties. We consider in particular the special cases of semi-diagonal and semi-trace cubic forms, the latter being a new generalization of the former.
Geometrical properties are then used to classify non-singular cubic pairs (A,V) over the separable closure of F, where we may suppose that A is a matrix algebra. Then we compute the automorphism group of such cubic pairs (which, in fact, is either the group of order three or the product of two such groups) and by means of its first Galois cohomology set we deduce a complete classification of non-singular cubic pairs over the ground field itself. By more computational methods we also give a complete classification of singular cubic pairs.
As an application we deduce in particular that the cubic form associated to a cubic pair (A,V) with A a division algebra is always semi-trace; and it is semi-diagonal if the ground field contains a primitive cube root of unity. Using a result of D. Haile and J.-P. Tignol we prove moreover that such a cubic form determines the algebra up to (anti-)isomorphism.
School:Université catholique de Louvain
Source Type:Master's Thesis
Keywords:formes de degré supérieur à 2 courbes géométrie algébrique
Date of Publication:12/07/2007