I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-archimedean valued field, and the action is $z\rightarrow qz$. There is a natural projection map: $\pi:G^{an}_{m,k}\rightarrow \Gamma$ and we define $e:=\pi(1)$. Then $\Gamma$ is a connected, non singular, separated and proper rigid space over $k$ of dimension $1$(lemma 5.1.1).

In theorem 5.1.4, by Riemann-Roch like in elliptic curves, we have the equation: $y^2+\lambda_1x^3+\lambda_2xy+\lambda_3x^2+\lambda_4y+\lambda_5x+\lambda_6=0$

So here are the questions(in the proof of theorem 5.1.4)

(1)How to prove $k[x,y]=\bigcup_{n\geq0}L(n[e])$ from dim$L(n[e])=n$ for each $n\geq1$?

(2)On page 92, the authurs say that the stalk is analytic local ring like $l\{Z_1,...,Z_n\}/I$ for some ideal $I$ where $l$ is the residue field of the stalk. But I don't know what the authurs mean in the proof of of theorem 5.1.4 "The syalk $O_E$ is isomorphic to the analytic local ring $k{s}$ for some local parameter $s$".

Maybe the questions are easy, but I still don't know why, thanks for your answers!

I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-archimedean valued field, and the action is $z\rightarrow qz$. There is a natural projection map: $\pi:G^{an}_{m,k}\rightarrow \Gamma$ and we define $e:=\pi(1)$. Then $\Gamma$ is a connected, non singular, separated and proper rigid space over $k$ of dimension $1$(lemma 5.1.1).

In theorem 5.1.4, by Riemann-Roch like in elliptic curves, we have the equation: $y^2+\lambda_1x^3+\lambda_2xy+\lambda_3x^2+\lambda_4y+\lambda_5x+\lambda_6=0$

So here are the questions(in the proof of theorem 5.1.4)

(1)How to prove $k[x,y]=\bigcup_{n\geq0}L(n[e])$ from dim$L(n[e])=n$ for each $n\geq1$?

(2)On page 92, the authurs say that the stalk is analytic local ring like $l\{Z_1,...,Z_n\}/I$ for some ideal $I$ where $l$ is the residue field of the stalk. But I don't know what the authurs mean in the proof of of theorem 5.1.4 "The syalk $O_E$ is isomorphic to the analytic local ring $k\{s\}$ for some local parameter $s$".

Maybe the questions are easy, but I still don't know why, thanks for your answers!

I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-archimedean valued field, and the action is $z\rightarrow qz$. There is a natural projection map: $\pi:G^{an}_{m,k}\rightarrow \Gamma$ and we define $e:=\pi(1)$. Then $\Gamma$ is a connected, non singular, separated and proper rigid space over $k$ of dimension $1$(lemma 5.1.1).

In theorem 5.1.4, by Riemann-Roch like in elliptic curves, we have the equation: $y^2+\lambda_1x^3+\lambda_2xy+\lambda_3x^2+\lambda_4y+\lambda_5x+\lambda_6=0$

So here are the questions(in the proof of theorem 5.1.4)

(1)How to prove $k[x,y]=\bigcup_{n\geq0}L(n[e])$ from dim$L(n[e])=n$ for each $n\geq1$?

(2)How to prove the curve $E$ with function field $k(x,y)$ has genus $1$ from dim$L(n[e])=n$ for each $n\geq1$? This is dual to this question, but now $E$ and $\Gamma$ are not the same curve.

(3)On page 92, the authurs say that the stalk is analytic local ring like $l\{Z_1,...,Z_n\}/I$ for some ideal $I$ where $l$ is the residue field of the stalk. But I don't know what the authurs mean in the proof of of theorem 5.1.4 "The syalk $O_E$ is isomorphic to the analytic local ring $k{s}$ for some local parameter $s$".

Maybe the questions are easy, but I still don't know why, thanks for your answers!

I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-archimedean valued field, and the action is $z\rightarrow qz$. There is a natural projection map: $\pi:G^{an}_{m,k}\rightarrow \Gamma$ and we define $e:=\pi(1)$. Then $\Gamma$ is a connected, non singular, separated and proper rigid space over $k$ of dimension $1$(lemma 5.1.1).

In theorem 5.1.4, by Riemann-Roch like in elliptic curves, we have the equation: $y^2+\lambda_1x^3+\lambda_2xy+\lambda_3x^2+\lambda_4y+\lambda_5x+\lambda_6=0$

So here are the questions(in the proof of theorem 5.1.4)

(1)How to prove $k[x,y]=\bigcup_{n\geq0}L(n[e])$ from dim$L(n[e])=n$ for each $n\geq1$?

(2)On page 92, the authurs say that the stalk is analytic local ring like $l\{Z_1,...,Z_n\}/I$ for some ideal $I$ where $l$ is the residue field of the stalk. But I don't know what the authurs mean in the proof of of theorem 5.1.4 "The syalk $O_E$ is isomorphic to the analytic local ring $k{s}$ for some local parameter $s$".

Maybe the questions are easy, but I still don't know why, thanks for your answers!