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Let $E$ be an elliptic curve over $\mathbb{Q}$. For a prime $p$, let $\mathcal{E}_p$ denote its Neron model over $\mathbb{Z}_p$. Also, let $\Phi_p(E)$ denote the component group of $\mathcal{E}_p$.

The structure of $\Phi_p(E)$ is well-known, and I want to study it when $E$ has multiplicative reduction at $p$. First, if $E$ has split multiplicative reduction at $p$, then it is known that $\Phi_p(E) \simeq \mathbb{Z}/{n\mathbb{Z}}$, where $n$ is the (normalized) $p$-adic valuation of the discriminant of $E$. (I will admit this fact.) Next, if $E$ has non-split multiplicative reduction at $p$, then $\Phi_p(E) \simeq \mathbb{Z}/{m\mathbb{Z}}$, where $m=1$ or $2$ such that $m \equiv n \pmod 2$.

Here is my question: Suppose that $E$ has non-split multiplicative reduction at $p$. As above, we use the same notation ($m$, $n$, $\Phi_p(E)$ etc). We know the following.

  1. There is a unramified quadratic extension $L/{\mathbb{Q}_p}$ such that $E/{\mathcal{O}_L}$ has split multiplicative reduction.

  2. Neron model does not change under etale base change.

If the above two are correct, then the component group $\Phi_p(E)$ can be computed using the Neron model of $E/L$, where $E$ has split multiplicative reduction. Since the $p$-adic valuation does not change under the unramified extension, the component group of the Neron model of $E/L$ is isomorphic to $\mathbb{Z}/{n\mathbb{Z}}$. Thus, $\Phi_p(E)$ is also isomorphic to $\mathbb{Z}/{n\mathbb{Z}}$, which is not true if $n>2$.

I don't know where my argument fails. Please correct me!

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The Galois group of $L/{\mathbb Q_p}$ acts on the component group over $L$ and the component group over $\mathbb Q_p$ should be the subgroup of elements fixed by the Galois group.

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