On the semi‐stability degree for abelian varieties

Abstract

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