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 $L(\Lambda_M)$, which is an important class of explicit function fields with applications in number theory and coding theory. Motivated by Arakelian and Quoos' classification of $L(\Lambda_M)$ with an irreducible quadratic modulus, we provide a complete characterization of the cyclotomic function field $L(\Lambda_M)$ 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 $L(\Lambda_{x^2})$ if and only if it satisfies the following three conditions: (i) $\mathcal{F}$ has a subgroup $G$ isomorphic to the direct product $(\mathbb{F}_q,+) \times \mathbb{F}_q^*$; (ii) its genus is $g(\mathcal{F}) = 1 + q(q-3)/2$; and (iii) the cardinality of $\mathbb{F}_q$-rational places is exactly $q+1$.