Does torsor of an elliptic curve extend to torsor of its Neron model? Let $(S,\eta,s)$ be spectrum of a discrete valuation ring $R$. Let $E$ be an elliptic curve over $\eta$. Let $\mathcal{E}$ be the Neron model of $E$.
Is there a concrete example of an $E$-torsor (smooth genus one curve over $K$) that does not extends to an $\mathcal{E}$-torsor over $S$?
(I am not even sure in the case when $E$ has good reduction, or $\mathcal{E}$ is smooth over $S$.)
 A: Any non-trivial $E$-torsor over $\mathbb{Q}_p$ will give you an example. Let me be more precise.
Let $E$ be an elliptic curve over $K=\mathrm{Frac}(R)$. Assume that $E$ has good reduction over $R$. Let $\mathcal{E}$ be its Neron model. If $X$ is an $E$-torsor over $K$ which extends to an $\mathcal{E}$-torsor $\mathcal{X}$ over $R$, then $X$ has    good reduction over $R$, as $\mathcal{X}$ is a smooth proper model for $X$ over $R$.
For certain residue fields, this will force the torsor $X$ to be split. For example, if $R=\mathbb{Z}_p$.


Theorem. Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $X$ be an $E$-torsor   over $\mathbb{Q}_p$.
    If  $X$ has good reduction over $\mathbb{Z}_p$, then $X$ is split.
Proof.
    It suffices to show that $X(\mathbb{Q}_p)$ is non-empty. Let $\mathcal{X}\to Spec R$ be a smooth proper model for $X$ over $R =\mathbb{Z}_p$. To show that  $X(\mathbb{Q}_p)$ is non-empty, it suffices to show that $\mathcal{X}(\mathbb{F}_p)$ is non-empty (Hensel). However, $\mathcal{X}_0:= \mathcal{X}_{\mathbb{F}_p}$ is a smooth proper geometrically connected genus one curve over $\mathbb{F}_p$, and such curves have an $\mathbb{F}_p$-point by the Hasse-Weil theorem. QED


Thus, it suffices to write down an elliptic curve over $\mathbb{Q}_p$ and any non-trivial $E$-torsor over $\mathbb{Q}_p$.
