uniformizer on elliptic curve in Silverman's book

I'd like to ask remark 1.1 in chapter 2 of Silverman's "Arithmetic of elliptic curves".

Let $$K$$ be a field and $$C$$ be a curve with a smooth point $$P$$, after proving $$\bar{K}[C]_P$$ is a discrete valuation ring in proposition 1.1, Silverman mentions that if $$P\in C(K)$$, then $$K(C)$$ contains uniformizers for $$P$$. In another word, there are uniformizers that are defined over K at $$P$$.

I want to know how to prove this and if there are some references of algebraic curves in Silverman's style in chapter II.

Thanks.

• I don't typically think about non-algebraically closed fields, but I think the idea is that if $P$ is defined over $K$, then evaluation is defined, so we have a surjective map $\mathcal O_{C,P} \to K$ given by $f \mapsto f(P)$. Its kernel is a maximal ideal, any generator of which is a uniformizer. Dec 2, 2020 at 2:00
• @TabesBridges. Does $\mathcal O_{C,P}=\bar{K}[C]_p$? I have studied some algebraic geometry, but when I see this chapter of Silverman, I am somewhat confused.
– user832207
Dec 2, 2020 at 2:11
• $\mathcal O_{C,P}$ is the local ring of $C$ at $P$. Whether this cleanly fits into Silverman's exposition I'm not sure. Dec 2, 2020 at 2:44
• This should basically be covered by this question. Dec 2, 2020 at 6:22

The short answer: Let $$P = (x_P,y_P)$$. Since $$P$$ is a smooth point on $$C$$, $$\left.\displaystyle \frac{dy}{dx}\right|_P$$ is well defined on $$C$$; note that the value may well be $$\infty$$. Then either $$\left.\displaystyle \frac{dy}{dx}\right|_P = 0$$ or $$\left.\displaystyle \frac{dy}{dx}\right|_P \neq 0$$ (the latter case includes the case $$\left.\displaystyle \frac{dy}{dx}\right|_P = \infty$$.)
If $$\left.\displaystyle \frac{dy}{dx}\right|_P = 0$$, then $$x-x_P$$ is a uniformizer. Otherwise $$y-y_P$$ is a uniformizer. I won't prove this for you because proving this on your own is a necessary step towards being able to understand the rest of the book, and so I suggest you do so.
• It seems that you've assumed $C$ is a plane curve in your answer. Is this sufficient to prove it in general? I can imagine trying to reduce to this case using projection, but it seems like this could fail if $K = \mathbb{F}_p$. Dec 5, 2020 at 1:40