Minimal Trivializing Isogenies of $\mathbb{G}_m$-gerbes over Abelian Varieties and Period-Index Problem

Abstract

For an abelian variety $X$ and $α\in Br(X)$, we propose a new invariance $Ind_{SH}(α)$ that refines the known period index relations. It is closely related to the geometry of $\mathcal{X}$, the $\mathbb{G}_m$-gerbe over $X$ that corresponds to $α$: we study the minimal trivializing isogenies for $\mathcal{X}$ via its $μ_n$-lifts and the $1-$twisted semi-homogeneous vector bundles on $\mathcal{X}$. As an application, we show that the period index conjecture holds true for products of elliptic curves of any dimension.