Let $E_q$ be the rigid analytic Tate elliptic curve over a complete algebraically closed non-archimedean field $K$ of mixed characteristic $(0,p)$, with parameter $q\in K^{\times}$ with $|q|<1$.

The unit point $e\in E_q(K)$, the projection of $1\in \mathbf{G}_m^{an}(K)$ to $E_q(K)$ under the quotient map $$\mathbf{G}_m^{an}\to \mathbf{G}_m^{an}/q^{\mathbf{Z}}\simeq E_q$$ defines an analytic line bundle $\mathcal{L} := \mathcal{O}(e)$.

Can we give a

geometricdescriptionof $\mathbf{P}(\mathcal{L})$ of the form $(\mathbf{G}_m^{an} \times_{\text{Sp}(K)} \mathbf{A}^1_K)/q^{\mathbf{Z}}$ for some (which one?) action of $q^{\mathbf{Z}}$ on $\mathbf{A}^1_K$?

My guess is, calling $z$ the coordinate on the affine line, to let $q$ act on $z$ as $q^{-1}z$. The resulting quotient $(\mathbf{G}_m^{an} \times_{\text{Sp}(K)} \mathbf{A}^1_K)/q^{\mathbf{Z}}$ should be the geometric line bundle on $E_q$ defined by either $\mathcal{L}$ or maybe $\mathcal{L}^{\otimes 3}$. I'm only saying this to give a sense of what I'm looking for. The action I propose could be wrong (or there may not be such a description)

**Reference on rigid Tate curves:**

Fresnel-van der Put, *Rigid Analytic Geometry and its Applications*, Ch. V, $\S$5.1.