19 problems

Goldring–Koskivirta's cone conjecture. The two cones coincide:

The saturated zip-cone conjecture. One has

Let $G$ be a connected reductive group over $\boldsymbol{F}_p$, let $\mu$ be a cocharacter of $G$, and let $r$ be a representation of $G$. The associated Griffiths-Hodge bundle is…

Cone conjecture. Under these assumptions, one has

G-zip encoding conjecture. The cone ${\mathcal C}(\overline{\mathbb{F}}_p)$ of weights admitting mod-$p$ Siegel automorphic forms is encoded by the stack of $G$-zips.

Finite generation conjecture. The ring $R_{\mathsf{zip}}$ is a finitely generated $k$-algebra; equivalently, the stack \mathop{\text{G-\tt ZipFlag}}\nolimits^\mu is a Mori drea…

Let $S_K$ be a Hodge-type Shimura variety, let ${\mathcal C}_K$ be the saturation of the set of characters whose associated automorphic vector bundles on $S_K$ have nonzero degree-…

Let $G$ be the reductive group over $\mathbb{F}_p$ attached to a Hodge-type Shimura variety, let $\mu$ be its Hodge cocharacter, and let $\mathop{\mathrm{Zip}}\nolimits_G^\mu$ be t…

Assume that $G$ is split over $\mathbb{F}_q$. Let $Y$ be the flag space associated with the cocharacter datum $(G,\mu)$, with flag strata $Y_w$ indexed by $w\in W$, and let…

Let $(G,\mu)$ be a cocharacter datum over $\mathbb{F}_q$, and let $(X,\zeta)$ satisfy the paper's stated assumptions. Let $Y$ be the flag space of $X$, stratified by flag strata…

The cone conjecture. Under these assumptions,

The cone conjecture. For any Hodge-type Shimura variety, one has

Let $k$ be a field, let $G$ be a reductive group over a finite field, let $\mu\colon\mathbb{G}_{\mathrm{m},k}\to G_k$ be a cocharacter, and let $(X,\zeta)$ consist of a $k$-scheme…

Let $G$ be a reductive group over a finite field $\mathbb{F}_q$, let $k=\overline{\mathbb{F}}_q$, and let $\mu\colon\mathbb{G}_{\mathrm{m},k}\to G_k$ be a cocharacter. Let…

Let $k$ be the ground field and let $R_{\rm zip}$ be the $k$-algebra of functions associated with the zip datum. Hilbert finite-generation conjecture. The $k$-algebra $R_{\rm zip}$…

Let $C_{\operatorname{GS}}$ denote the cone of weights arising from the geometric automorphic forms considered in the source, and let $C_{\operatorname{zip}}$ be the cone of charac…

Let $R_{\rm zip}$ be the graded algebra … where $\mathscr V(\lambda)$ is the automorphic vector bundle associated with $\lambda$. Finite generation conjecture. The…

Inert Hilbert modular stratum conjecture. If $w\in W$ has length two, then

Global sections cone conjecture. Assume that the zip datum ${\mathcal Z}$ is of connected-Hodge-type, that for every connected component $X^\circ\subset X$ the map…