Definition of good(bad) reductioness of elliptic curve by not using immersions in $\mathbb{P}^2$

143 Views Asked by Bumbble Comm At 13 Apr 2026 - 4:12

Sorry for my bad English.

Let $K$ be a number field, and $E$ be an elliptic curve over $K$,and $\frak{p}$ be a finite place of $K$.

Then I know the definition of good(bad) reductionness at $\frak{p}$ of $E$ by using minimal Weierstrass equation.

But I think it looks like dependent on immersion in projective place $\mathbb{P}^2$.

So I want to know is there another equivalent definition of good(bad) reductionness by not using immersion in $\mathbb{P}^2$.

Also is there relation Neron model?

I have basic knowledge about algebraic geometry such as Hartshorne, but almost don't know about elliptic curve.

There are 1 best solutions below

Bumbble Comm On 05 Jun 2022 - 2:08 BEST ANSWER

(Not enough reputation to comment)

To define reduction, what you need is a good notion of integral model. Good/bad reduction at $p$ is then whether the fiber of the integral model at $\mathbb{Z}_p$ is still an elliptic curve.
Minimal Weierstrass model is a good integral model for the case of elliptic curves. You are not happy with its definition because it depends on an embedding in $\mathbb{P}^2$. So one way to reframe your question is whether there is an embedding-free definition of good integral model.
Neron model is one such embedding-free definition. It's defined by a universal property (and so by abstract nonsense it is unique if it exists), but it's not clear if exists or not. For the case of elliptic curves it does exist and coincides with the minimal Weierstrass model.