超楕円関数体の類数について

Let F=GF(p) be a finite prime field of characteristic p≠2. Let K=F(x,y) be a hyperelliptic function field over F defined by an equation y2=xn+a (a≠O, a∈F), where n denotes an odd number such that n>1 and p∤n. Let h be the class number of K and g the genus of K. Then, we have proved that h=p+1 if n=3 and p≡2 mod 3. (〔4〕, Theorem 1 (i)). This particular fact can be generally expressed as follows; Given n, there exists an integer c such that h=pg+ 1 whenever p≡c mod n. In this note, it is shown that this generalization is true in the particular case of n=5 and of n=7