Trilinear alternating forms and related CMLs and GECs

The classification of trivectors(trilinear alternating forms) depends essentially on the dimension n of the base space. This classification seems to be a difficult problem (unlike in the bilinear case). For n\leq ۸ there exist finitely many trivector classes under the action of the general linear group GL(n). The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from \bar{K}(the algebraic closure of K) to K. In this paper, we are interested in the classification of trivectors of an eight dimensional vector space over a finite field of characteristic ۳, % K=\mathbb{F}_{۳^{m}}. We obtain a ۳۱ inequivalent trivectors, ۲۰ of which are full rank. Having its motivation in the theory of the generalized elliptic curves and commutative moufang loop, this research studies the case of the forms over the ۳ elements field. We use a transfer theorem providing a one-to-one correspondence between the classes of trilinear alternating forms of rank ۸ over a finite field with ۳ elements \mathbb{F}_{۳} and the rank ۹ class ۲ Hall generalized elliptic curves (GECs) of ۳-order ۹ and commutative moufang loop (CMLs). We derive a classification and explicit descriptions of the ۳۱ Hall GECs whose rank and ۳-order both equal ۹ and the number of order ۳^{۹}-CMLs. 

کلیدواژه ها:

Commutative moufang loops ، Generalized elliptic curves ، Trivectors ، Classification