The $(p,t,a)$-inertial groups as finite monodromy groups

Abstract

Silverberg and Zarhin introduced the notion of a $(p,t,a)$-inertial group in the hope of having a group theoretic characterization of the finite groups that appear as finite monodromy groups -- the groups that represent the local obstruction to semi-stable reduction -- of abelian varieties in fixed dimension $t+a$. In this text, we provide a positive answer to their question, that is, every $(p,t,a)$-inertial group is the finite monodromy group of an abelian variety in dimension $t+a$. To prove this, we show a structure theorem on the rational group algebra $\mathbf{Q}[G]$ of ramification groups, refining a theorem of Serre and generalizing results on $p$-groups of Roquette and Ford.