On class numbers of hyperelliptic function fields with Hasse-Witt invariant zero
Let \(q=p^ r\), \(K=GF(q)\), and \(a\in K^ *\), where p is a prime number, and r and g are positive integers. Denote by ord(a) the order of a in the cyclic group \(K^ *\) and put \(I(n,a)=(q^ n-1)/ord(a)\) for \(n=1,2,... \). In this note the L-function L(u) and the class number \((=L(1))\) of some kind of hyperelliptic function field A having Hasse-Witt invariant zero are determined. Main results are stated as follows. Theorem. (i) If \(A=K(x,y)\) is defined by \(y^ 2=x^{2g+1}+a\) and \(p\equiv -1\) (mod 4g\(+2)\), then \[ L(u)=(1+qu^ 2)^ g\quad if^ 2 \nmid r\quad,\quad L(u)=\frac{(1-c^ k u^ k)^{(2g+1)/k}}{1-cu}\quad if^ 2 | r\quad, \] where \(c=(-1)^{I(1,a)} (-p)^{r/2}\) and \(k=Min\{n\geq 1\); \(2g+1 | I(n,a)\}.\) (ii) If \(A=K(x,y)\) is defined by \(y^ 2=x(x^{2g}+a)\) and \(p\equiv -1\) or \(1+2g\) (mod 4g), then \[ L(u)=(1-cu^ k)^{2g/k}\quad, \] where \(c=(-1)^{I(k,a)/2g} (-p)^{kr/2}\) and \(k=Min\{n\geq 1\); 2g \(| I(n,a)\) and 2 \(| nr\}.\) The proof of the theorem is based on the congruence relation between L(u) and the Hasse-Witt matrix of A and on the Riemann hypothesis in the theory of algebraic function fields.
scientific article; zbMATH DE number 3995830