Special fibers of Néron models and wild ramification (Q2706831)

Let \(K\) be a field with a discrete valuation. Let \({\mathcal O}_K\) denote the ring of integers of \(K\). Let \(k\) be the residue field of \({\mathcal O}_K\), assumed to be algebraically closed of characteristic \(p>0\). Let \(G/K\) be a commutative group scheme with Néron model \({\mathcal G}/{\mathcal O}_K\). Let \({\mathcal G}_k/k\) be the special fiber of \({\mathcal G}/{\mathcal O}_K\), and let \({\mathcal G}_k^0/k\) denote the connected component of 0 in \({\mathcal G}_k\). The group of components of \({\mathcal G}\) is the abelian group \(\Phi(G): ={\mathcal G}_k/{\mathcal G}_k^0\). We say that \(G/K\) has split reduction if the extension NEWLINE\[NEWLINE0\to{\mathcal G}_k^0(k)\to {\mathcal G}_k(k) @>c>> \Phi(G)\to 0NEWLINE\]NEWLINE is split. The core of this article is a detailed study of the splitting properties of elliptic curves and of norm tori and their duals, with applications to abelian varieties with potentially purely multiplicative reduction. In all cases studied, we find that there exists a constant \(c\) depending only on the dimension of \(G\) such that, if \(G\) has totally non split reduction, then the Swan conductor of \(G/K\) is positive and bounded by \(c\). We also find that there is a constant \(d\), depending only on the dimension of \(G\), such that \(G_M/M\) has split reduction for any tame extension \(M/K\) of degree greater than \(d\). Clearly this suggests the possibility that similar statements may hold for more general tori and abelian varieties.