Remark 55.18.2 (Improving the bound). Results in the literature suggest that one can improve the bound given in the statement of Theorem 55.18.1. For example, in [DM] it is shown that semistable reduction of $C$ and its Jacobian are the same thing if the residue field is perfect and presumably this is true for general residue fields as well. For an abelian variety we have semistable reduction if the action of Galois on the $\ell $-torsion is trivial for any $\ell \geq 3$ not equal to the residue characteristic. Thus we can presumably choose $\ell = 5$ in the formula (55.18.0.1) for $B_ g$ (but the proof would take a lot more work; if we ever need this we will make a precise statement and provide a proof here).

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