level structure $\lambda$ **completely determines** $\psi$.

Let $\hat V:=V\cup\{\infty\}$ for a new symbol $\infty$. We add a “point at infinity” via the open embedding $${{\mathbb{G}}}_{a,K}\into{{\mathbb{P}}}^1_K,\,\,t\mapsto (t:1),$$ and obtain a map $\hat\lambda\colon \hat V\to{{\mathbb{P}}}^1_K(K)$ from $\lambda$ by sending $\infty$ to $(1:0)$. The pair $({{\mathbb{P}}}^1_K,\lambda)$ is then a smooth stable $\hat V$-marked curve. One obtains a left action of the group $$G:=V\rtimes{{\mathbb{F}}}_q^\times$$ on ${{\mathbb{P}}}^1_K$ via the homomorphism $$\phi\colon G\to\PGL_2(K),\,\,(v,\xi)\mapsto\begin{pmatrix}
\xi& \lambda_v\\0 & 1
\end{pmatrix}.$$ This leads to the following definition:

\[smoothferndef\] A ***smooth** $V$-fern* over $K$ is a tuple $(C,\lambda,\phi)$ consisting of a smooth stable $\hat V$-marked curve $(C,\lambda)$ and a left $G$-action $$\phi\colon G\to \Aut_K(C),\,\,(v,\xi)\mapsto \phi_{v,\xi}$$ such that

1.  $\phi_{v,\xi}(\lambda_w)=\lambda_{\xi w+v}$ for all $w\in \hat V$, and

2.  there exists an isomorphism $C\stackrel\sim\to{{\mathbb{P}}}^1_K$ with $\lambda_0\mapsto(0:1)$ and $\lambda_\infty\mapsto(1:0)$ under which $\phi_{\xi}$ corresponds to $\begin{pmatrix}\xi &0\\0&1
    \end{pmatrix}\in\PGL_2(K)$ for all $\xi\in{{\mathbb{F}}}_q^\times$.

The upshot of the preceding discussion is that one can naturally associate a smooth $V$-fern over $K$ to a Drinfeld ${{\mathbb{F}}}_q[t]$-module of rank $n$ and level $(t)$ over $K$. In the above notation, this is given by $(\psi,\lambda)\mapsto \lambda\mapsto ({{\mathbb{P}}}^1_K,\hat\lambda,\phi)$. After passing to isomorphism classes, the notions are in fact equivalent.

We can generalize to the notion of a smooth $V$- fern over 