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.