Extending torsors over regular models of curves

Let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $k$ of characteristic $p>0$. Given a finite commutative group scheme $G$ over $K$ and a smooth projective curve $C$ over $K$ with a rational point, we study the extension of pointed fppf $G$-torsors over $C$ to pointed torsors over some $R$-regular model $\mathcal{C}$ of $C$. We first study this problem in the category of log schemes: given a finite flat $R$-group scheme $\mathcal{G}$, we prove that the data of a pointed $\mathcal{G}$-log torsor over $\mathcal{C}$ is equivalent to that of a morphism $\mathcal{G}^D \to \mathrm{Pic}^{log}_{\mathcal{C}/R}$, where $\mathcal{G}^D$ is the Cartier dual of $\mathcal{G}$ and $\mathrm{Pic}^{log}_{\mathcal{C}/R}$ the log Picard functor. Then, we deduce a criterion for the extension of torsors: it suffices to find a finite flat model of $G$ over $R$ for which a certain group scheme morphism to the Jacobian $J$ of $C$ extends to the N\'eron model of $J$. In this context, we compute the obstruction for the extended log torsor to come from an fppf one. In a second part, we generalize a result of Chiodo which gives a criterion for the $r$-torsion subgroup of the N\'eron model of $J$ to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give two detailed examples of extension of torsors when $C$ is a hyperelliptic curve defined over $\mathbb{Q}$, which will illustrates our techniques.