Igusa stacks product conjecture for minimal compactifications of Shimura varieties

Establish the existence and compatibility of Igusa stacks in the following sense: (1) extend the Hodge–Tate period map to the minimal compactification by constructing \pi_{HT}^\diamond: (Sh_{K^p}^\ast)^\diamond \to Fl_{[\mu]}^\diamond for any compact open K^p \leq G(\mathbb{A}^{\infty p}); and (2) construct, functorially in K^p, small v-stacks Igs_{K^p}^\ast, together with maps \overline{\pi}_{HT}: Igs_{K^p}^\ast \to Bun\,G and q_{K^p}: (Sh_{K^p}^\ast)^\diamond \to Igs_{K^p}^\ast, such that \overline{\pi}_{HT} is 0-truncated and the Cartesian diagram (Sh_{K^p}^\ast)^\diamond \cong Fl_{[\mu]}^\diamond \times_{Bun\,G} Igs_{K^p}^\ast holds.

Background

The paper’s construction of inscribed global Shimura varieties relies on a fiber-product description involving the Hodge–Tate period domain and stacks of G-bundles on the Fargues–Fontaine curve. This description depends on a conjectural product formula for Igusa stacks that refines the geometry of minimal compactifications via the Hodge–Tate period map.

The conjecture generalizes known results (proved in many PEL cases and compact Hodge type cases) and provides the stack-theoretic foundation needed to carry out moduli-theoretic constructions at infinite level, yielding torsor structures and tangent bundle computations consistent with the inscribed framework.