The Tamagawa Number of $A$ at $\ell$

In this section, the notation and hypothesis are as in Proposition 5.2. That proposition implies that the Tamagawa number $c_{A,\ell}=\char93 \Phi_{A,\ell}(\mathbb{F}_\ell)$ of $A$ at $\ell$ is coprime to $n$. In this section we use Remark 5.4 of [6] to prove that in fact $c_{A,\ell}=1$.

Let $\lambda$ be the prime of $K$ lying over $\ell$, and let $K_{\lambda}$ denote the completion of $K$ at $\lambda$, so $K_{\lambda}$ is totally and tamely ramified over $\mathbb{Q}_\ell$. Since

$$A_{K} \cong \ker(\Sigma: E_{K}^{\oplus n} \rightarrow E_{K}),$$

and $E_{K_\lambda}$ has good reduction, the geometric closed fiber of the Néron model of $A_{K_{\lambda}}$ is $A'_{\overline{k}}\cong \ker(\Sigma : E_{\overline{k}}^{\oplus n} \rightarrow E_{\overline{k}}).$ In the notation of [6], $\mu_n$ acts on $A'_{\overline{k}}$ by the action it induces by cyclically permuting the factors of $E_{\overline{k}}^{\oplus n}$. Thus $A_{\overline{k}}'(\overline{k})^{\mu_n}$ is the set of $\sum P_\sigma\otimes \sigma \in E(\overline{k})^{\oplus n}$ such that all $P_\sigma$ are equal and $\sum P_\sigma = 0$, i.e.,

$$A_{\overline{k}}'(\overline{k})^{\mu_n} \cong E(\overline{k})[n]\approx (\mathbb{Z}/n\mathbb{Z})^2.$$

Thus Remark 5.4 in [6] implies that $\Phi_{A,\ell}(\overline{k}) \approx E(\overline{k})[n]$. By Proposition 5.2, $\Phi_{A,\ell}(k)$ has no elements of order dividing $n$, so $\Phi_{A,\ell}(k)=0$.