Classification of function fields with class number three (Q2349350)

The aim of this paper is to give a complete classification of congruence function fields \(K/{\mathbb F}_q\) with class number \(h_K=3\). The case \(h_K=1\) was solved by \textit{R. E. MacRae} [J. Algebra 17, 243--261 (1971; Zbl 0212.53302)], \textit{M. L. Madan} and \textit{C. S. Queen} [Acta Arith. 20, 423--432 (1972; Zbl 0237.12007)] and \textit{J. R. C. Leitzel} et al. [J. Number Theory 7, 11--27 (1975; Zbl 0318.12009)] (up to one missing field). The case \(h_K=2\) was settled by Leitzel, Madan and Queen [loc. cit.] and \textit{D. Le Brigand} [Finite Fields Appl. 2, No. 2, 153--172 (1996; Zbl 0931.11051), in: Arithmetic, geometry, and coding theory. Proceedings of the international conference held at CIRM, Luminy, France, 1993. Berlin: Walter de Gruyter. 105--126 (1996; Zbl 0872.11049)]. The quadratic algebraic function fields of class number three were classified by \textit{A. Picone} [Discrete Math. 312, No. 3, 637--646 (2012; Zbl 1276.11188)]. In this work the nonquadratic algebraic function fields with \(h_K=3\) are classified. The main result is presented in several cases. Theorem 4.2 classifies the function fields such that \(K/{\mathbb F}_q\) is nonhyperelliptic of genus \(3\). There exist exactly \(4\) such fields. In Theorem 4.3 it is proved that there are \(14\) nonhyperelliptic function fields of genus \(4\). In Theorem 4.5 are considered the nonhyperelliptic function fields of genus \(5\). There exist \(5\) such fields. Finally, it is shown the non-existence of such nonhyperelliptic function fields of genus \(6\).