(A) generation of class fields over function field

Abstract

Let $k$ be a global function field with a fixed prime divisor $\infty$. Let $A$ be the ring of integers outside $\infty$. D.Hayes generated three Class Fields, Hilbert Class Field $H_A$, Normalizing Field $\tilde H_A$, and $K_{\frak m} = \tilde H_A(\Lambda_{\frak m})$ over $k$, using elliptic $A$-module of rank 1 of generic characteristic i and $sgn$-normalized elliptic $A$-module([1],[2]). In this paper, we extend the above result to the case that $A$ is replaced by an order $R$ of $A$. Similary we generate three Class Fields, Hilbert Class Field of $R$, $H_R$, Normalizing Field $\tilde H_R$, and $\tilde K_{\frak m} = \tilde H_R(\Lambda_{\frak m})$ over $k$. We also identify them by using Class Field Theory as follows, $$\begin{array}{ccc}H_R&\longleftrightarrow &J_R \\\tilde H_R&\longleftrightarrow & \tilde J_R \\K_{\frak m} &\longleftrightarrow & \tilde J_{\frak m}\end{array}$$ where $J_R = k^{\times}\cdot\pi_{\infty}^{\Bbb Z}\cdot U_R$ ; \ $\tilde{J_R} = k^{\times}\cdot\pi_{\infty}^{\Bbb Z}\cdot \tilde U_R$ ; \ $\tilde J_{\frak m} = k^{\times}\cdot \pi_{\infty}^{\Bbb Z} \cdot\tilde U_{\frak m}$.