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Some remarks on the number of points on elliptic curves over finite prime field

Published online by Cambridge University Press:  17 April 2009

Saiying He  and
J. McLaughlin
Saiying He
Affiliation:
Trinity College, 300 Summit Street, Hartford, CT 06106–3100, United States of America, e-mail: [email protected]
J. McLaughlin
Affiliation:
Mathematics Department, West Chester University, West Chester, PA 19383, e-mail: [email protected]
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Let p ≥ 5 be a prime and for a, b ε p, let Ea, b denote the elliptic curve over p with equation y2 = x3 + ax + b. As usual define the trace of Frobenius ap, a, b by

We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums and for primes p in various congruence classes.

As an example of our results, we prove the following: Let p ≡ 5 (mod 6) be prime and let b ε *p. Then

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 1 , February 2007 , pp. 135 - 149
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Berndt, B.C., Evans, R.J., and Williams, K.S., Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley and Sons, Inc., New York, 1998).Google Scholar
[2]Birch, B.J., ‘How the number of points of an elliptic curve over a fixed prime field varies.’, J. London Math. Soc. 43 (1968), 57–60.CrossRefGoogle Scholar
[3]Blake, I.F., Seroussi, G. and Smart, N.P., Elliptic curves in cryptography, London Mathematical Society Lecture Note Series 265, (Reprint of the 1999 original) (Cambridge University Press, Cambridge, 2000).Google Scholar
[4]Hasse, H., ‘Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F.K. Schmidtschen Kongruenzzetafunktionen in gewissen eliptischen Fällen’, in Vorläufige Mitteilung, Nachr. Ges. Wiss. Göttingen I, Math.-phys. Kl. Fachgr. I Math. Nr. 42, 1933, pp. 253–262.Google Scholar
[5]Ireland, K. and Rosen, M., A classical introduction to modern number theory, Graduate Texts in Mathematics 84, (Second edition) (Springer-Verlag, New York, 1990).CrossRefGoogle Scholar
[6]Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia of Mathematics and its Applications 20, (Second edition) (Cambridge University Press, Cambridge, 1997).Google Scholar
[7]Michel, P., ‘Rang moyen de families de courbes elliptiques et lois de Sato-Tate’, Monatsh. Math. 120 (1995), 127–136.CrossRefGoogle Scholar