Primitive elements for \(p\)-divisible groups (Q2409030)

Let \(\mathcal G\) be a finite flat group scheme. Using Raynaud's theory of Haar measures of [\textit{M. Raynaud}, Bull. Soc. Math. Fr. 102, 241--280 (1974; Zbl 0325.14020)], the authors define a closed subscheme \(\mathcal G^\times\hookrightarrow\mathcal G\) by decreeing its ideal to be the line bundle of invariant measures on the Cartier dual of \(\mathcal G\). Moreover, if \(\mathcal G\) is a Barsotti-Tate group of level \(n\) for a prime number \(p\), the authors define the closed subscheme \(\mathcal G^{prim}\hookrightarrow\mathcal G\) of ``primitive elements'' to be the inverse image of \(\mathcal G[p]^\times\) under the canonical epimorphism \(\mathcal G\twoheadrightarrow\mathcal G[p]\). This concept paves the way for a new approach to integral models \(\mathfrak X\) of Shimura varieties with level \(\Gamma_1(p^n)\)-structure, simply by looking at the scheme \(\mathcal A[p^n]^{prim}\) where \(\mathcal A\) is the universal abelian \(g\)-fold (possibly with additional structure). These schemes are not necessarily normal, but they are still interesting counterweights to the more classical concept of moduli spaces of points ``with exact order \(p^n\)'', which seems to be limited to settings in which one can work with one-dimensional formal groups, as in [\textit{V. G. Drinfel'd}, Math. USSR, Sb. 23, 561--592 (1976; Zbl 0321.14014); translation from Mat. Sb., n. Ser. 94(136), 594--627 (1974)] or [\textit{N. M. Katz} and \textit{B. Mazur}, Arithmetic moduli of elliptic curves. Princeton, NJ: Princeton University Press (1985; Zbl 0576.14026)].