On Class Numbers of Algebraic Function Fields defined by y^2=x^2g+1+ax with g=2 and 3

Let F=GF(P) be a finite prime field of characteristic P≠2. Let K=F(x, y) be an algebraic Function field over F defined by an equation y2=x2g+1+ax (a≠0, a∈F), where g means a positive integer. Let h be the class number of K Moreover, let us denots by xd a character of order d of the cyclic group F= F-{0}. Then, it is shown that