Minimal models for drinfeld modules of rank 2 with complex multiplication

Abstract

In this paper, we study on the minimal models of Drinfeld module of rank 2. Let F be a separable extension of k = $F_q(T).$ In the first, we show that if the class number $h(O_F)$ is greater than 1, then there exists a Drinfeld module over F which does not have a global minimal model over F. Let K be a imaginary quadratic extension of k and H be the Hilbert class field of $Ο_k$. Let φ be a Drinfeld module defined over H of rank 2 with complex multiplication by $Ο_k$. We prove that if q is odd and p(T) is a monic irreducible element in $F_q[T]$ of degree prime to q-1, then there exists a unique k-module which has a global minimal model over k(j(φ)).