A Note on Cyclotomic Function Fields with Quadratic Modulus

A longstanding and important problem in algebraic geometry is the characterization of algebraic function fields. In this paper, we focus on the characterization problem for cyclotomic function field $L({\mathrm{\Lambda }}_M)$, which is an important class of explicit function fields with applications in number theory and coding theory. Motivated by Arakelian and Quoos’ classification of $L({\mathrm{\Lambda }}_M)$ with an irreducible quadratic modulus, we provide a complete characterization of the cyclotomic function field $L({\mathrm{\Lambda }}_M)$ with modulus $M=x^2$. More precisely, we prove that a function field $\mathcal{F}$ over ${\mathbb{F}}_q$ is ${\mathbb{F}}_q$-isomorphic to $L({\mathrm{\Lambda }}_{x^2})$ if and only if it satisfies the following three conditions: (i) $\mathcal{F}$ has a subgroup $G$ isomorphic to the direct product $({\mathbb{F}}_q,+)\times {\mathbb{F}}_q^{\ast }$; (ii) its genus is $g(\mathcal{F})=1+q(q-3)\mathrm{/}2$; and (iii) the cardinality of ${\mathbb{F}}_q$-rational places is exactly $q+1$.