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 $\mathcal{L} := \mathcal{O}(e)$.



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> Can we give a **geometric** **description** of $\mathbf{P}(\mathcal{L})$ 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 $\mathbf{A}^1\_K$?
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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.



Geometric line bundles on the Tate curve
I think the action is $q\star z = -q^{-1} z$, so that you get the "correct" basic theta function. See p. 128 in Fresnel-van der Put or Roquette's book.


