Number of points on certain hyperelliptic curves defined over finite fields

作者：Anuradha N^*

来源：Finite Fields and Their Applications, 2008, 14(2): 314-328.

DOI：10.1016/j.ffa.2006.12.007

• Jacobsthal sums
• Jacobsthal-Whiteman sums
• curves
• finite fields
• zeta functions

For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums phi l(nu), psi l(nu), and psi 2l (nu), and the Jacobsthal-Whiteman sums phi ln (nu) and phi 2ln (nu), over finite fields F(q) such that q = p(alpha) equivalent to 1 (mod 2l). These results are obtained only in terms of q and l. We apply these results pertaining to the Jacobsthal sums, to determine, for each integer n >= 1, the exact number of F(q)n-rational points on the projective hyperelliptic curves aY(2)Z(e-2) = bX(e) + cZ(e) (abc not equal 0) (for e = l, 2l), and aY(2)Z(l-1) = X(bX(l) + cZ(l)) (abc not equal 0), defined over such finite fields F(q). As a consequence, we obtain the exact form of the zeta-functions for these three classes of curves defined over F(q), as rational functions in the variable t, for all distinct cases that arise for the coefficients a. b, c. Further, we determine the exact cases for the coefficients a, b, c, for each class of curves, for which the corresponding non-singular models are maximal (or minimal) over F(q).

• 出版日期2008-4
• 单位常州工学院