Ramification of a finite flat group scheme over a local field
Let \(K\) be a complete discrete valuation field of mixed characteristic \((0,p)\), whose residue field need not be perfect, and \(\mathcal O_K\) its valuation ring. The author investigates ramification of a finite flat group scheme \(\mathcal G\) over \(\mathcal O_K\) following the results of \textit{A. Abbes} and \textit{T. Saito} [Am. J. Math. 124, 879--920 (2002; Zbl 1084.11064)], in particular the conductor \(c(\mathcal G)\) as introduced in Proposition 6.4 of that paper. Supposing that \(\mathcal G\) is killed by \(p^n\), it is proved that \[ c(\mathcal G) \leq ne + e/(p-1), \] where \(e\) is the absolute ramification index of \(K\). This result allows a generalization of a theorem of \textit{J.-M. Fontaine} [Invent. Math. 81, 515--538 (1985; Zbl 0612.14043)]. If \(\mathcal G\) is the \(p^n\)-torsion part of an elliptic curve, the exact value of \(c(\mathcal G)\) is given, depending on the elliptic curve being ordinary, fake ordinary, or truly supersingular.
scientific article; zbMATH DE number 5072770