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The characterization of elliptic curves over finite fields

Part of: Curves Finite geometry and special incidence structures

Published online by Cambridge University Press:  09 April 2009

J. W. P. Hirschfeld  and
J. F. Voloch
J. W. P. Hirschfeld
Affiliation:
Mathematics Division, University of SussexBrighton BN1 9QH, United Kingdom
J. F. Voloch
Affiliation:
I.M.P.A., Est. D. Castorina 110, Rio de Janeiro 22460, Brazil
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Abstract

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In a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

MSC classification

Secondary: 51E15: Affine and projective planes 14H25: Arithmetic ground fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 2 , October 1988 , pp. 275 - 286
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bierbrauer, J., ‘Sets of maximal size in the plane of order eight’, preprint.Google Scholar
[2]De Groote, R. and Hirschfeld, J. W. P., ‘The number of points on an elliptic cubic curve over a finite field’, European J. Combin. 1 (1980), 327–333.CrossRefGoogle Scholar
[3]Hartshorne, R., Algebraic geometry (Springer, 1977).CrossRefGoogle Scholar
[4]Hirschfeld, J. W. P., Projective geometries over finite fields (Oxford University Press, 1979).Google Scholar
[5]Hirschfeld, J. W. P., ‘Maximum sets in finite projective spaces’, Surveys in combinatorics, pp. 55–76 (London Math. Soc. Lecture Note Series 82, 1983).CrossRefGoogle Scholar
[6]Hirschfeld, J. W. P., Finite projective spaces of three dimensions (Oxford University Press, 1985).Google Scholar
[7]Keedwell, A. D., ‘When is a (k, n)-arc of PG(2, q) embeddable in a unique algebraic plane curve of order n?’, Rend Mat. 12 (1979), 397–410.Google Scholar
[8]Keedwell, A. D., ‘A theorem concerning the embedding of graphic arcs in algebraic plane curves’, preprint.Google Scholar
[9]Keedwell, A. D., ‘Simple constructions for elliptic cubic curves with specified small numbers of points’, preprint.Google Scholar
[10]Mason, J. R. M., ‘A class of ((pn – pm)(pn – 1), pn – pm)-arcs in PG(2, pn)’, Geom. Dedicata 15 (1984), 355–361.Google Scholar
[11]Yazdi, M. Oraee, The classification of (k; 3)-arcs over the Galois field of order five (Thesis, University of Sussex, 1986).Google Scholar
[12]Raguso, G. and Rella, L., ‘Graphic arcs of order 5, 6 embeddable in algebraic plane curves of the same order’, Mitt. Math. Sem. Giessen 166 (1984), 167–176.Google Scholar
[13]Raguso, G. and Rella, L., ‘On the graphic arcs embeddable in an algebraic plane curves’, Mitt. Math. Sem. Giessen 169 (1985), 45–53.Google Scholar
[14]Raguso, G. and Rella, L., ‘Sugli archi grafici di ordine 3 di un piano di Galois di caratteristica 2’, Boll. Un. Mat. Ital. 4-D (1985), 161–166.Google Scholar
[15]Schoof, R. J., ‘Nonsingular plane cubic curves over finite fields’ J. Combin. Theory, Ser. A 46 (1987), 183–211.CrossRefGoogle Scholar
[16]Serre, J.-P., Nombres de points des courbes algébriques sur Fq', Séminaire de Théorie des Nombres de Bordeaux, exposé no. 22, 1983).Google Scholar
[17]Serre, J.-P., ‘Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini’, C. R. Acad. Sci. Paris Sér. I 296 (1983), 397–402.Google Scholar
[18]Stöhr, K.-O. and Voloch, J. F., ‘Weierstrass points and curves over finite fields’, Proc. London Math. Soc. 52 (1986), 1–19.CrossRefGoogle Scholar
[19]Scafati, M. Tallini, ‘Graphic curves on a Galois plane’, Atti del Convegno di Geometria Combinatoria e sue Applicazioni, Università di Perugia, 1970, pp. 413–419 (Università di Perugia, 1971).Google Scholar
[20]Thas, J. A., ‘Elementary proofs of two fundamental theorems of B. Segre without using the Hasse-Weil theorem’, J. Combin. Theory Ser. A 34 (1983), 381–384.CrossRefGoogle Scholar
[21]Thas, J. A., ‘Complete arcs and algebraic curves in PG(2, q)’, perprint.Google Scholar
[22]Voloch, J. F., ‘On the completeness of certain plane arcs’, European J. Combin., to appear.Google Scholar
[23]Voloch, J. F., ‘Arcs in projective planes over prime fields’, J. Geom., to appear.Google Scholar
[24]Waterhouse, W. G., ‘Abelian varieties over finite fields’, Ann. Sci. École Norm. Sup. 2 (1969), 521–560.CrossRefGoogle Scholar
[25]Yasin, A. L., Cubic arcs in the projective plane of order eight (Thesis, University of Sussex, 1986).Google Scholar