An alternative way to classify some Generalized Elliptic Curves and their isotopic loops

Lucien Bénéteau; M. Abou Hashish

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 2, page 237-255
  • ISSN: 0010-2628

Abstract

top
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 α ( n , m , K ) the cardinal number of the sets of representatives. We establish that α ( 5 , 2 , K ) 5 whenever each field-element is quadratic; moreover α ( 5 , 2 , 𝔽 3 ) = 6 and α ( 6 , 2 , 𝔽 3 ) 13 . We obtained a transfer theorem providing a one-to-one correspondence between the classes from AT ( n , m , 𝔽 3 ) and the rank n + 1 class 2 Hall GECs of 3 -order n + m . Now α ( 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 ( 𝔽 3 ) .

How to cite

top

Bénéteau, Lucien, and Hashish, M. Abou. "An alternative way to classify some Generalized Elliptic Curves and their isotopic loops." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 237-255. <http://eudml.org/doc/249353>.

@article{Bénéteau2004,
abstract = {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: $(x*x^\{\prime \},y,z)=(x,y,z)*(x^\{\prime \},y,z)$. 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 $\alpha (5,2,\mathbb \{F\}_\{3\})=6$ and $\alpha (6,2,\mathbb \{F\}_\{3\})\ge 13$. We obtained a transfer theorem providing a one-to-one correspondence between the classes from $\operatorname\{AT\}(n,m,\mathbb \{F\}_\{3\})$ 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 $G_\{2\}(\mathbb \{F\}_\{3\})$.},
author = {Bénéteau, Lucien, Hashish, M. Abou},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves; extended triple systems; alternate trilinear mappings; totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves},
language = {eng},
number = {2},
pages = {237-255},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An alternative way to classify some Generalized Elliptic Curves and their isotopic loops},
url = {http://eudml.org/doc/249353},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Bénéteau, Lucien
AU - Hashish, M. Abou
TI - An alternative way to classify some Generalized Elliptic Curves and their isotopic loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 2
SP - 237
EP - 255
AB - 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: $(x*x^{\prime },y,z)=(x,y,z)*(x^{\prime },y,z)$. 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 $\alpha (5,2,\mathbb {F}_{3})=6$ and $\alpha (6,2,\mathbb {F}_{3})\ge 13$. We obtained a transfer theorem providing a one-to-one correspondence between the classes from $\operatorname{AT}(n,m,\mathbb {F}_{3})$ 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 $G_{2}(\mathbb {F}_{3})$.
LA - eng
KW - totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves; extended triple systems; alternate trilinear mappings; totally symmetric quasigroups; terentropic quasigroups; commutative Moufang loops; generalized elliptic curves
UR - http://eudml.org/doc/249353
ER -

References

top
  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. (1981) Zbl0482.20044MR0624133
  3. Bénéteau L., Extended triple systems: geometric motivations and algebraic constructions, Discrete Math. 208/209 (1999), 31-47. (1999) MR1725518
  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. MR0794742
  5. Bénéteau L., Lacaze J., Symplectic trilinear form and related designs and quasigroups, Comm. Algebra 16 (5) (1988), 1035-1051. (1988) MR0926336
  6. Bénéteau L., Razafimanantsoa G., Boucles de Moufang k-nilpotentes minimales, C.R. Acad. Sci. Paris, Série I 306 (1988), 743-746. (1988) MR0948765
  7. Buekenhout F., Generalized elliptic cubic curves, Part 1, Finite Geometries, (2001), 35-48. Zbl1014.51003MR2060755
  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. Zbl0719.20036MR1125806
  9. Cohen A., Helminck A., Trilinear alternating forms on a vector space of dimension 7 , Comm. Algebra 16.1 (1988), 1-25. (1988) Zbl0646.15019MR0921939
  10. Djokovic D.Z., Classification of 3 -vectors of a real 8 -dimensional vector space, Linear and multilinear algebra (1983), 3-39. (1983) MR0691457
  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. (1990) Zbl0718.17028MR1055009
  12. Gurewitch G.B., Foundations of the Theory of Algebraic Invariants, P. Noordhoff LTD, Groningen, Netherlands, 1964. MR0183733
  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. Zbl0786.51009MR1249252
  14. Kepka T., Němec P., Commutative Moufang loops and distributive groupoids of small orders, Czechoslovak Math. J. 31 (106) (1981), 633-669. (1981) MR0631607
  15. Kepka T., Structure of triabelian quasigroups, Comment. Math. Univ. Carolinae 17 (1976), 229-240. (1976) Zbl0338.20097MR0407182
  16. Koblitz N., A course in Number Theory and Cryptography, Second Edition, New-York, Springer-Verlag, 1994. Zbl0819.11001MR1302169
  17. Manin Yu.I., Cubic Forms, Algebra, Geometry, Arithmetic, North-Holland, Amsterdam, London, 1974. Zbl0582.14010MR0833513
  18. Němec P., Commutative Moufang loops corresponding to linear quasigroups, Comment. Math. Univ. Carolinae 29 (1988), 303-308. (1988) MR0957400
  19. Noui L., Formes multilinéaires alternées, Thèse de troisième cycle, Université de Montpellier II, 1995. Zbl0831.15017
  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. 112, (1988), 357-368. MR0975369
  22. Schoof R., Counting points on elliptic curves over finite fields, Journal de Théorie des nombres de Bordeaux VII, (1995), 219-254. Zbl0852.11073MR1413578
  23. Schouten J.A., Klassifizierung der alternierender Grössen dritten Grades in 7 Dimensionen, Rend. Circ. Nat. di Palermo 55 (1931), 137-156. (1931) 
  24. Schwenk J., A classification of abelian quasigroups, Rend. Math. Appl. (7) 15 (2) (1995), 161-172. (1995) Zbl0831.05015MR1339239
  25. Smith J.D.H., Finite equationally complete entropic quasigroups, Contribution to General Algebra, Proc. Klagenfurt Conf., 1978 pp.345-355. Zbl0412.20070MR0537430
  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. MR0504529
  27. Westwick R., Real trivectors of rank seven, Linear and Multilinear Algebra (1980), 183-204. (1980) Zbl0439.15014MR0630147

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.