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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: Saiying.He@trincoll.edu
J. McLaughlin
Affiliation:
Mathematics Department, West Chester University, West Chester, PA 19383, e-mail: jmclaughl@wcupa.edu
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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

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