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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 $S$.)

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  • $\begingroup$ Which Neron model? The smooth, but not necessarily proper, one, or the proper regular, but not necessarily smooth, one. $\endgroup$ – anon Oct 31 '19 at 18:08
  • $\begingroup$ @anon The smooth one? $\endgroup$ – Qixiao Oct 31 '19 at 23:26
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    $\begingroup$ This is not really a concrete example, but the condition of an $E$ torsor $X$ extending to a $\mathscr E$ torsor is equivalent to whether $X$ has any points over the fraction field of the strict henselization of R. See Neron Models, by Bosch, Lütkebohmert, and Raynaud, section 6.5, Corollaries 3 and 4 and the intervening discussion. No good reduction hypotheses are needed. So, any torsor which is nontrivial over the fraction field of the strict henselization gives an example, and conversely all examples must be of this form. $\endgroup$ – Aaron Landesman Nov 3 '19 at 5:28
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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 $\mathbb{Q}_p$.

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