Algebraic function fields with small class number

It is proved that there is no congruence function field of genus 4 over GF(2) which has no prime of degree less than 4 and precisely one prime of degree 4. This shows the nonexistence of function fields of genus 4 with class number one and gives an example of an isogeny class of abelian varieties which contains no jacobian. It is shown that, up to isomorphism, there are two congruence function fields of genus 3 with class number one. It follows that there are seven nonisomorphic function fields of genus different from zero with class number one. Congruence function fields with class number 2 are fully classified. Finally, it is proved that there are eight imaginary quadratic function fields $\text{F}\text{K(x)}$ for which the integral closure of Kx] in F has class number 2.