Isogeny of Drinfeld module

Question

Let $\phi$ be a Drinfeld module over a function field $K$. It is known that if the rank of $\phi$ is 1, then it is isomorphic over $K$ to one defined over $O_K$. Is it true that if the rank of $\phi$ is arbitrary, then it is isogenous over $K$ to one defined over $O_K$?

Answer 1

Yes, this always holds. In fact if $A$ is the $\mathbb{F}_q$-algebra of functions on a curve $X/\mathbb{F}_q$ which are regular away from some point $\infty \in X$, then we can write $A = \mathbb{F}_q[x_1, \dots, x_m]$. A Drinfeld $A$-module $\phi$ defined over a global function field $K$ (for simplicity, assume $A \subseteq K$), is given by an $\mathbb{F}_q$-algebra homomorphism $\phi : A \to K[\tau]$, where $\tau$ is the $q$-th power Frobenius endomorphism.

Suppose that for each $i$, $$ \phi_{x_i} = x_i + a_{i,1} \tau + \cdots + a_{i,\ell_i} \tau^{\ell_i}, \quad a_{i,j} \in K. $$ Pick $u \in O_K \setminus \{0 \}$ so that for each $i$, $j$, we have $u a_{i,j} \in O_K$. Then conjugating $\phi$ by $u$, we have an isomorphic (over $K$) Drinfeld module $\phi' : A \to K[\tau]$ so that $$ \phi'_{x_i} := u^{-1} \phi_{x_i} u = x_i + u^{q-1} a_{i,1} \tau + \cdots + u^{q^{\ell_i}-1} a_{i,\ell_i} \tau^{\ell_i}, $$ which is defined over $O_K$.

On another note, it may not be possible to do this so that $\phi'$ has everywhere good reduction. See Chapter 7 of Goss's book "Basic Structures of Function Field Arithmetic" for more on this.