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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 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 $\mathbf{A}^1_K$?

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.

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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.

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    $\begingroup$ P.S. You may also check out section 9.6 in the lecture notes: achinger.impan.pl/rigid/notes.pdf , but keep in mind that they are far from being in a stable form. $\endgroup$ Commented Jun 12, 2021 at 18:26

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