Splitting properties of the reduction of semi-abelian varieties

Abstract: Let $K$ be a complete discrete valuation field. Let $\mathcal{O}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\geq1$. Let $G/K$ be a semi-abelian variety. Let $\mathcal{G}/\mathcal{O}_K$ be its N\'eron model. The special fiber $\mathcal{G}_k/k$ is an extension of the identity component $\mathcal{G}_k^0/k$ by the group of components $\Phi(G)$. We say that $G/K$ has split reduction if this extension is split. Whereas $G/K$ has always split reduction if $p=1$ we prove that it is no longer the case if $p>1$ even if $G/K$ is tamely ramified. If $J/K$ is the Jacobian variety of a smooth proper and geometrically connected curve $C/K$ of genus $g$, we prove that for any tamely ramified extension $M/K$ of degree greater than a constant, depending on $g$ only, $J_M/M$ has split reduction. This answers some questions of Liu and Lorenzini.