On the degree of semi-stable reduction

Abstract

For an abelian variety $A$ over a number field we study bounds depending only on the dimension of $A$ for the minimal degree $d(A)$ of a field extension over which $A$ acquires semi-stable reduction. We first compute $d(A)$ in terms of the cardinalities of the finite monodromy groups of $A$ which leads to a bound on $d(A)$ in terms of the classical Minkowski bound. We then show this bound is tight up to its $2$-part by considering $p$-adic coverings of the local points of a universal abelian scheme.