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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 $E_K$.

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    $\begingroup$ 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. $\endgroup$
    – BCnrd
    Commented May 17, 2010 at 15:31
  • $\begingroup$ By the way, the "N-gon" description of the special fiber is not quite right; that is only for the geometric special fiber. $\endgroup$
    – BCnrd
    Commented May 18, 2010 at 5:40

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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 $N$.

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  • $\begingroup$ Note that, to consolidate with BCnrd's comment, I am assuming that your family has regular total space. If not, it has at worst ordinary double point singularities in the special fiber, which can be resolved by blowing up, which will increase the number of components (in a way which depends upon the analytic local ring of at each singular point). $\endgroup$
    – Pete L. Clark
    Commented May 17, 2010 at 15:40

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