# Article

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Keywords:
totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves; extended triple systems; alternate trilinear mappings
Summary:
The Generalized Elliptic Curves $(\operatorname{GECs})$ 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 $\operatorname{GEC}$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $\operatorname{GEC}$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(\operatorname{CML})$. If in addition $x^2=x$, we have Hall $\operatorname{GECs}$ and $(Q,*)$ is an exponent $3$ $\operatorname{CML}$. Any finite terentropic $\operatorname{GEC}$ 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 $\operatorname{GECs}$ (including just three non-entropic $\operatorname{GECs}$). In class $2$ $\operatorname{CMLs}$ 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 $\operatorname{AT}(n,m,K)$ 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 $\operatorname{AT}(n,m,\Bbb F_{3})$ and the rank $n+1$ class $2$ Hall $\operatorname{GECs}$ of $3$-order $n+m$. Now $\alpha (7,1,\operatorname{GF}(3^s))=11$ for any $s$. We derive a complete classification and explicit descriptions of the eleven Hall $\operatorname{GECs}$ whose rank and $3$-order both equal $8$. One of these has for automorphism group some extension of the Chevalley group $G_{2}(\Bbb F_{3})$.
References:
[1] Abou Hashish M.: Applications trilinéaires alternées et courbes cubiques elliptiques généralisées classifications et utilisations cryptographiques. Thèse de Doctorat, no. 687, Institut National des Sciences Appliquées de Toulouse, 2003.
[2] Bénéteau L.: Ordre minimum des boucles de Moufang commutatives de classe $2$ (resp. $3$). Ann. Fac. Sci. Toulouse Math. (5) 3 (1981), 75-88. MR 0624133 | Zbl 0482.20044
[3] Bénéteau L.: Extended triple systems: geometric motivations and algebraic constructions. Discrete Math. 208/209 (1999), 31-47. MR 1725518
[4] Bénéteau L., Kepka P.: Quasigroupes trimédiaux et boucles de Moufang commutatives libres. C.R. Acad. Sci. Paris, t. 300, Série I, no. 12 (1985), 377-380. MR 0794742
[5] Bénéteau L., Lacaze J.: Symplectic trilinear form and related designs and quasigroups. Comm. Algebra 16 (5) (1988), 1035-1051. MR 0926336
[6] Bénéteau L., Razafimanantsoa G.: Boucles de Moufang k-nilpotentes minimales. C.R. Acad. Sci. Paris, Série I 306 (1988), 743-746. MR 0948765
[7] Buekenhout F.: Generalized elliptic cubic curves. Part 1, Finite Geometries, (2001), 35-48. MR 2060755 | Zbl 1014.51003
[8] Chein O., Pflugfelder H.O., Smith J.D.H.: Quasigroups and Loops; Theory and Applications. Sigma Series in Pure Mathematics, vol. 8, Heldermann, Berlin, 1990. MR 1125806 | Zbl 0719.20036
[9] Cohen A., Helminck A.: Trilinear alternating forms on a vector space of dimension $7$. Comm. Algebra 16.1 (1988), 1-25. MR 0921939 | Zbl 0646.15019
[10] Djokovic D.Z.: Classification of $3$-vectors of a real $8$-dimensional vector space. Linear and multilinear algebra (1983), 3-39. MR 0691457
[11] Griess R.L., Jr.: A Moufang loop, the exceptional Jordan algebra, and a cubic form in $27$ variables. J. Algebra 131 1 (1990), 281-295. MR 1055009 | Zbl 0718.17028
[12] Gurewitch G.B.: Foundations of the Theory of Algebraic Invariants. P. Noordhoff LTD, Groningen, Netherlands, 1964. MR 0183733
[13] Keedwell A.D.: More simple constructions for elliptic cubic curves with small numbers of points. Pure Math. Appl. Ser. A, Vol. 3, No. 3-4, (1992), 199-214. MR 1249252 | Zbl 0786.51009
[14] Kepka T., Němec P.: Commutative Moufang loops and distributive groupoids of small orders. Czechoslovak Math. J. 31 (106) (1981), 633-669. MR 0631607
[15] Kepka T.: Structure of triabelian quasigroups. Comment. Math. Univ. Carolinae 17 (1976), 229-240. MR 0407182 | Zbl 0338.20097
[16] Koblitz N.: A course in Number Theory and Cryptography. Second Edition, New-York, Springer-Verlag, 1994. MR 1302169 | Zbl 0819.11001
[17] Manin Yu.I.: Cubic Forms, Algebra, Geometry, Arithmetic. North-Holland, Amsterdam, London, 1974. MR 0833513 | Zbl 0582.14010
[18] Němec P.: Commutative Moufang loops corresponding to linear quasigroups. Comment. Math. Univ. Carolinae 29 (1988), 303-308. MR 0957400
[19] Noui L.: Formes multilinéaires alternées. Thèse de troisième cycle, Université de Montpellier II, 1995. Zbl 0831.15017
[20] Razafimanantsoa G.: La k-nilpotence minimale dans les boucles de Moufong commutatives; classification partielle des applications trilinéaires alternées. Thèse no. 3511, Univ. Toulouse III, 1988.
[21] Revoy Ph.: Fomes trilinéaires alternées de rang $7$. Bull. Sci. Math. $2^e$112, (1988), 357-368. MR 0975369
[22] Schoof R.: Counting points on elliptic curves over finite fields. Journal de Théorie des nombres de Bordeaux VII, (1995), 219-254. MR 1413578 | Zbl 0852.11073
[23] Schouten J.A.: Klassifizierung der alternierender Grössen dritten Grades in $7$ Dimensionen. Rend. Circ. Nat. di Palermo 55 (1931), 137-156.
[24] Schwenk J.: A classification of abelian quasigroups. Rend. Math. Appl. (7) 15 (2) (1995), 161-172. MR 1339239 | Zbl 0831.05015
[25] Smith J.D.H.: Finite equationally complete entropic quasigroups. Contribution to General Algebra, Proc. Klagenfurt Conf., 1978 pp.345-355. MR 0537430 | Zbl 0412.20070
[26] Vinberg E.B., Elasvili A.G.: Classification of trivectors of a nine-dimensional space. Trudy Sem. Vekt. Tenz. Analizu, no. XVIII, (1978), 197-223. MR 0504529
[27] Westwick R.: Real trivectors of rank seven. Linear and Multilinear Algebra (1980), 183-204. MR 0630147 | Zbl 0439.15014

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