Bounds of Ideal Class Numbers of Real Quadratic Function Fields, by Kunpeng WANG and XianKe ZHANG

Abstract: The theory of continued fractions of functions $ \sqrt D $ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt D)$ over $k={\bf F}_q(T)$. For five series of real quadratic function fields $K$, the bounds of $h(D)$ are given more explicitly, e.g., if $\ D=F^2+c,$ \mbox{}\hspace{0.1cm} then $\ h(D)\geq \mbox{deg}F /\mbox{deg} P;$ \hspace{0.1cm} if $D=(SG)^2+cS,\ $ then $\ h(D)\geq \mbox{deg}S / \mbox{deg} P;\; $ if $D=(A^m+a)^2+A,\ $ then $\ h(D)\geq \mbox{deg}A / \mbox{deg} P, \; $ where $P$ is irreducible polynomial splitting in $K,\; c\in {\bf F}_q$ is any constant. In addition, six types of quadratic function fields are found to have ideal class numbers bounded and bigger than one. \\ {\bf keywords:} quadratic function field, ideal class number, continued fractions of functions

Kunpeng WANG and XianKe ZHANG <xianke@tsinghua.edu.cn , xzhang@math.tsinghua.edu.cn>