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 $S$.)



Does torsor of an elliptic curve extend to torsor of its Neron model?
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 $R=\mathbb{Z}\_p$.



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


