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I am looking for an "easy-to-understand" reference for Neron Models. Specifically if I have a semi-stable family of elliptic curves over $Spec {O}_K$ , with generic fibre $E_K$ and special fibre $E_k$ , then $E_k$ is an $N$-gon of $\mathbb{P}^1$'s. In this context, what is the Neron model of $E_K$? I guess what I am asking is for a geometric description of the special fibre of the Neron model for $E_K$.

flag	asked May 17 2010 at 15:20 New_to_MO 11●1

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Neron model is always the smooth locus of minimal regular proper model; these matters are discussed (with references) in the later part of Q. Liu's book "Algebraic geometry and arithmetic curves". The sst model you have above may be non-regular, for example. If you begin with a regular sst proper flat model and make a ramified base change then regularity is generally lost but is regained by blow-ups; that's why even in mult. reduction case the structure of the component group of geometric special fiber is affected by ramified base change even though relative identity component is not. – BCnrd May 17 2010 at 15:31

By the way, the "N-gon" description of the special fiber is not quite right; that is only for the geometric special fiber. – BCnrd May 18 2010 at 5:40

Pete L. Clark · Answer 1 · 2010-05-17 15:34:42Z UTC

The place to look for this is Chapter 4 ("The Neron Model") of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves, specifically Theorem 4.6.1: the Neron model of an elliptic curve is obtained by removing the singular points from the minimal regular proper model.

Thus in your case the connected component is a rational curve with two points removed: as a group it is $\mathbb{G}_m$, the multiplicative group. The component group here is cyclic of order $N$.

Semi-stable model and Neron model for family of elliptic curves

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