Uncertain about Uniformizing Elements of Elliptic Curves.

Question

I am following a subject on Elliptic Curves and have come accross the notion of a uniformizer. Wikipedia tells me that an element is a uniformizer of a Discrete Valuation Ring, if it generates the (only) maximal ideal. This seems sort of clear, but I have no idea how to apply it to elliptic curves. Consider the following question:

Let $k$ a field, $C: y^2=x$ a smooth curve in $\mathbb{A}^2$ and $P=(\alpha,\beta)$ a point in $C(k)$. Furthermore suppose that the characteristic of $k\neq 2$. Show that $x-\alpha$ is a uniformizing element of $P$ if and only if $P\neq (0,0)$.

Now this is not even intuitively clear to me. The ideal we want to look at is $(y-\beta,x-\alpha)$ I suppose, since this maps $k[x,y]/(y^2-x)$ to $0\in k$, but how do I show that $(y-\beta,x-\alpha)=(x-\alpha)$ iff $P\neq (0,0)$?

I also cannot find any information about such problems anywhere (I have the book Rational points on elliptic curves by Silvermann, but it has nothing about uniformizers).

I would appreciate some explanation (or a solution with an explanation so I can apply this to other problems) or a reference to a book which explains this to somebody who has not heard about Discrete Valuation Rings or Uniformizers before.

EDIT: This is still not clear to me, I tried finding info in the recommended book, but it still doesn't offer enough information. Could anybody be so helpful to explain how to find uniformizers for such functions?

Answer 1

As this is homework, I'll try not to say too much. Recall the definitions:

The local ring of $C$ at $P=(\alpha,\beta)\in C(k)$ is $k[C]_{\mathfrak{p}}$, where $k[C]=k[x,y]/(y^2-x)$ and $\mathfrak{p}=(x-\alpha,y-\beta)k[C]$. Its maximal ideal is $\mathfrak{m}_P=\mathfrak{p}k[C]_{\mathfrak{p}}$. A uniformizing element of $P$ is a generator of $\mathfrak{m}_P$.

Beyond this, the exercise requires no knowledge of DVR's, only some basic facts on local rings.

Hint 1:

Show that $y-\beta$ is a uniformizing element of $P$.

Hint 2:

If $y+\beta$ is a unit in the local ring, then $x-\alpha$ is a uniformizing element of $P$.

Hint 3:

Show that $y+\beta$ is a unit in the local ring if and only if $2\beta\neq0$.

This covers the 'if' part of the exercise. The 'only if' part should not be hard once you understand the 'if' part.

Answer 2

I'm attending the same course as you, and was trying to find the same answers when Google turned up your question. I just emailed the lecturer with this question, and he suggested Silverman's "The arithmetic of Elliptic Curves", in particular chapter II (without II.4: differentials), algorithm III.2.3, and propositions III.3.1 and III.3.4.