| Title:
|
An alternative way to classify some Generalized Elliptic Curves and their isotopic loops (English) |
| Author:
|
Bénéteau, L. |
| Author:
|
Hashish, M. Abou |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
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45 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
237-255 |
| . |
| Category:
|
math |
| . |
| 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 $G_{2}(\Bbb F_{3})$. (English) |
| Keyword:
|
totally symmetric quasigroups |
| Keyword:
|
terentropic quasigroups |
| Keyword:
|
commutative Moufang loops |
| Keyword:
|
generalized elliptic curves |
| Keyword:
|
extended triple systems |
| Keyword:
|
alternate trilinear mappings |
| MSC:
|
11G07 |
| MSC:
|
14H52 |
| MSC:
|
20N05 |
| MSC:
|
46G25 |
| idZBL:
|
Zbl 1102.20052 |
| idMR:
|
MR2075272 |
| . |
| Date available:
|
2009-05-05T16:44:47Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119453 |
| . |
| Reference:
|
[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. |
| Reference:
|
[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. Zbl 0482.20044, MR 0624133 |
| Reference:
|
[3] Bénéteau L.: Extended triple systems: geometric motivations and algebraic constructions.Discrete Math. 208/209 (1999), 31-47. MR 1725518 |
| Reference:
|
[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 |
| Reference:
|
[5] Bénéteau L., Lacaze J.: Symplectic trilinear form and related designs and quasigroups.Comm. Algebra 16 (5) (1988), 1035-1051. MR 0926336 |
| Reference:
|
[6] Bénéteau L., Razafimanantsoa G.: Boucles de Moufang k-nilpotentes minimales.C.R. Acad. Sci. Paris, Série I 306 (1988), 743-746. MR 0948765 |
| Reference:
|
[7] Buekenhout F.: Generalized elliptic cubic curves.Part 1, Finite Geometries, (2001), 35-48. Zbl 1014.51003, MR 2060755 |
| Reference:
|
[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. Zbl 0719.20036, MR 1125806 |
| Reference:
|
[9] Cohen A., Helminck A.: Trilinear alternating forms on a vector space of dimension $7$.Comm. Algebra 16.1 (1988), 1-25. Zbl 0646.15019, MR 0921939 |
| Reference:
|
[10] Djokovic D.Z.: Classification of $3$-vectors of a real $8$-dimensional vector space.Linear and multilinear algebra (1983), 3-39. MR 0691457 |
| Reference:
|
[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. Zbl 0718.17028, MR 1055009 |
| Reference:
|
[12] Gurewitch G.B.: Foundations of the Theory of Algebraic Invariants.P. Noordhoff LTD, Groningen, Netherlands, 1964. MR 0183733 |
| Reference:
|
[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. Zbl 0786.51009, MR 1249252 |
| Reference:
|
[14] Kepka T., Němec P.: Commutative Moufang loops and distributive groupoids of small orders.Czechoslovak Math. J. 31 (106) (1981), 633-669. MR 0631607 |
| Reference:
|
[15] Kepka T.: Structure of triabelian quasigroups.Comment. Math. Univ. Carolinae 17 (1976), 229-240. Zbl 0338.20097, MR 0407182 |
| Reference:
|
[16] Koblitz N.: A course in Number Theory and Cryptography.Second Edition, New-York, Springer-Verlag, 1994. Zbl 0819.11001, MR 1302169 |
| Reference:
|
[17] Manin Yu.I.: Cubic Forms, Algebra, Geometry, Arithmetic.North-Holland, Amsterdam, London, 1974. Zbl 0582.14010, MR 0833513 |
| Reference:
|
[18] Němec P.: Commutative Moufang loops corresponding to linear quasigroups.Comment. Math. Univ. Carolinae 29 (1988), 303-308. MR 0957400 |
| Reference:
|
[19] Noui L.: Formes multilinéaires alternées.Thèse de troisième cycle, Université de Montpellier II, 1995. Zbl 0831.15017 |
| Reference:
|
[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. |
| Reference:
|
[21] Revoy Ph.: Fomes trilinéaires alternées de rang $7$.Bull. Sci. Math. $2^e$112, (1988), 357-368. MR 0975369 |
| Reference:
|
[22] Schoof R.: Counting points on elliptic curves over finite fields.Journal de Théorie des nombres de Bordeaux VII, (1995), 219-254. Zbl 0852.11073, MR 1413578 |
| Reference:
|
[23] Schouten J.A.: Klassifizierung der alternierender Grössen dritten Grades in $7$ Dimensionen.Rend. Circ. Nat. di Palermo 55 (1931), 137-156. |
| Reference:
|
[24] Schwenk J.: A classification of abelian quasigroups.Rend. Math. Appl. (7) 15 (2) (1995), 161-172. Zbl 0831.05015, MR 1339239 |
| Reference:
|
[25] Smith J.D.H.: Finite equationally complete entropic quasigroups.Contribution to General Algebra, Proc. Klagenfurt Conf., 1978 pp.345-355. Zbl 0412.20070, MR 0537430 |
| Reference:
|
[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 |
| Reference:
|
[27] Westwick R.: Real trivectors of rank seven.Linear and Multilinear Algebra (1980), 183-204. Zbl 0439.15014, MR 0630147 |
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