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

For an abelian variety $X$ and $\alpha \in Br(X)$, we propose a new invariance $In{d}_{SH}(\alpha )$ 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 $\alpha$: we study the minimal trivializing isogenies for $\mathcal{X}$ via its ${\mu }_n$-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.