Class Numbers of Real Quadratic Function Fields of Genus One

For a prime numberp, let pbe the finite field of cardinalitypandX=Xpa fixed indeterminate. We prove that for any natural numberN, there exist infinitely many pairs (p,K/ p(X)) of a prime numberpand a “real” quadratic extensionK/ p(X) for which the genus ofKis one and the class number of the integral closure of p[X] inKisN.