Number of points on certain hyperelliptic curves defined over finite fields

For odd primes p and l such that the order of p modulo l is even, we determine explicitly the Jacobsthal sums ϕl(v), ψl(v), and ψ2l(v), and the Jacobsthal–Whiteman sums and , over finite fields Fq such that . 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 Fqn-rational points on the projective hyperelliptic curves aY2Ze−2=bXe+cZe (abc≠0) (for e=l,2l), and aY2Zl−1=X(bXl+cZl) (abc≠0), defined over such finite fields Fq. As a consequence, we obtain the exact form of the ζ-functions for these three classes of curves defined over Fq, 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 Fq.