For odd primes
p and
l such that the order of
p modulo
l is even, we deter
mine explicitly the Jacobsthal sums
l(v),
ψl(v), and
ψ2l(v), and the Jacobsthal–Whiteman sums
miImageURL/B6WFM-4MM7TG6-1-S/0?wchp=dGLbVzW-zSkWz" alt="View the MathML source" title="View the MathML source" align="absbottom" border="0" height=17 width="38"/> and
miImageURL/B6WFM-4MM7TG6-1-T/0?wchp=dGLbVzW-zSkWz" alt="View the MathML source" title="View the MathML source" align="absbottom" border="0" height=17 width="41"/>, over finite fields
Fq such that
miImageURL/B6WFM-4MM7TG6-1-W/0?wchp=dGLbVzW-zSkWz" alt="View the MathML source" title="View the MathML source" align="absbottom" border="0" height=15 width="136"/>. These results are obtained only in terms of
q and
l. We apply these results pertaining to the Jacobsthal sums, to deter
mine, 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 deter
mine 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.