The zeta function of stacks of $G$-zips and truncated Barsotti-Tate groups

Throughout this article, let p be a prime number. Over a field k of characteristic p, the truncated Barsotti–Tate groups of level 1 (henceforth BT1) were first classified in [9]. The main examples of BT1s come from p-kernels A[p] of abelian varieties A over k. As such, these results (independently obtained) were used in [13] to define a stratification on the moduli space of polarised abelian varieties. In [11] the first step was made towards generalising this to Shimura varieties of PEL type; this corresponds to Barsotti–Tate groups of level 1 with the action of a fixed semisimple Fp-algebra and/or a polarisation. The classification of these BT1s with extra structure over an algebraically closed field k̄ turned out to be related to the Weyl group of an associated reductive group over k̄. These BT1s with extra structure were then generalised in [12] to so-called F -zips, that generalise the linear algebra objects that arise when looking at the Dieudonné modules corresponding to BT1s. Over an algebraically closed field the classification of these F -zips was also