M. Abou Hashish, L. B\'en\'eteau
An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Comment.Math.Univ.Carolinae 45,2 (2004) 237-255.

Abstract:The Generalized Elliptic Curves $(GECs)$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of ``points'' from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $GEC$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(CML)$. If in addition $x^2=x$, we have Hall $GECs$ and $(Q,*)$ is an exponent $3$ $CML$. Any finite terentropic $GEC$ admits a direct decomposition in primary components and only the $3$-component may eventually be non entropic, in which case its order is at least $81$. It turns out that there are fifteen order $81$ terentropic $GECs$ (including just three non-entropic $GECs$). In class $2$ $CMLs$ the associator enjoys some pseudo-linearity: $(x*x',y,z)=(x,y,z)*(x',y,z)$. We are thus led to searching representatives in the set $AT(n,m,K)$ of image-rank $m$ alternate trilinear mappings from $(V(n,K))^3$ to $V(m,K)$ up to changes of basis in these $K$-vector spaces. Denote by $\alpha (n,m,K)$ the cardinal number of the sets of representatives. We establish that $\alpha (5,2,K)\le 5$ whenever each field-element is quadratic; moreover $\alpha (5,2,\Bbb F_{3})=6$ and $\alpha (6,2,\Bbb F_{3})\geq 13$. We obtained a transfer theorem providing a one-to-one correspondence between the classes from $AT(n,m,\Bbb F_{3})$ and the rank $n+1$ class $2$ Hall $GECs$ of $3$-order $n+m$. Now $\alpha (7,1,GF(3^s))=11$ for any $s$. We derive a complete classification and explicit descriptions of the eleven Hall $GECs$ whose rank and $3$-order both equal $8$. One of these has for automorphism group some extension of the Chevalley group $G_{2}(\Bbb F_{3})$.

Keywords: totally symmetric quasigroups, terentropic quasigroups, commutative Moufang loops, generalized elliptic curves, extended triple systems, alternate trilinear mappings
AMS Subject Classification: 20N05, 14H52, 46G25