164 problems

Zilber--Pink conjecture. If $V$ is an irreducible Hodge generic subvariety of $S$, then the intersection of $V$ with the special subvarieties of $S$ having codimension greater than…

For a Shimura datum of the relevant parahoric type, with maximal parahoric level at an odd prime $p$, let $M^{\mathrm{loc}}$ be the corresponding canonical local model over the rin…

Let $(G,X)$ and $(G',X')$ be Shimura data whose underlying groups are pure inner forms of one another, and let $\operatorname{Ig}(G,X)$ and $\operatorname{Ig}(G',X')$ denote the as…

Let $\mathcal{S}$ be a split unitary Shimura variety of signature $(g,g)$ at a split prime $p$, and let $\mathcal{S}_{\mathrm{basic}}$ denote its basic locus. Outside the degenerat…

Let $\Gamma\backslash X^+$ be an arithmetic variety, let $V$ be an algebraic subvariety, and let $\Sigma$ be the set of special points of $\Gamma\backslash X^+$ that belong to $V$.…

Let $\mathcal{A}_g$ be the moduli space of principally polarized complex abelian varieties of dimension $g$, let $T_g$ be the Torelli locus, and let $T_g^o$ be its open Torelli loc…

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum with reflex field $E$, let $G$ be its associated reductive group, and let $\mu$ be the Hodge cocharacter. Let $(r_\mu,V)$ be the re…

Let $G$ be a reductive group over $\mathbb{Q}$ admitting a Shimura datum $(G,X)$, and let $(\operatorname{Sh}_{K,\mathbb{C}})_K$ be the associated inverse system of Shimura varieti…

Assume the setup of the paper: $\Pi_0$ and $\Pi_1$ are the relevant representations under consideration, $F$ is the CM field, $E$ is their coefficient field, and the notation…

Dual EL-shellability conjecture. The poset $\widehat{\operatorname{Cox}(\mu)^K}$ is dual EL-shellable.

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

Let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}^{\mathrm{ord}}$ be a map from a smooth connected variety, let $x\in X(\mathbb{F})$, and let…

Let $p>0$, let $g$ be a positive integer, and let $x=\left[\left(A_x,\lambda_x\right)\right]$ be a $\overline{\mathbb{F}}_p$-point of the moduli space $\mathcal{A}_g$ of $g$-dimens…

Let $S$ be a Shimura variety and let $V$ be an irreducible subvariety of $S$. André–Pink–Zannier conjecture. Suppose that the intersection of $V$ with a single Hecke orbit in $S$ i…

Structural decomposition conjecture.

Let $\Gamma\backslash X^+$ be a Shimura variety, let $V$ be an algebraic subvariety of it, and let $x\in X^+$. Write ${\mathcal{H}}_{\Gamma\backslash X^+}(x)$ for the generalised H…

The saturated zip-cone conjecture. One has

Consider a Shimura datum $(G,X)$ with $G=\operatorname{Res}_{F/\mathbb{Q}}H$, where $H$ is a connected reductive group over a totally real field $F$. Let…

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$ over $\mathbb{C}$, let $Z\subset\mathcal{A}_g$ be an irreducible algebraic subva…

Let $n \in 2\mathbb{Z}_{\geq 1}$. Write $\Gamma^n_{\mathrm{eis}} \setminus \mathbb{R} H^n$ for the non-arithmetic hyperbolic orbifold $M({\mathscr{L}}_{\mathrm{eis}}^n,\alpha_0)$,…

Mazur's principle. Suppose that $m$ is not pseudo-Eisenstein and that $N^r=0$ on a representation space for $\overline{\rho}_m$. Then there is a level $U'$ containing $U$, isomorph…

Let $Sh_{K_p}$ be a PEL Shimura variety with standard parahoric level subgroup $K_p$, and let … be the map sending a point to the $c3$-conjugacy class of its associated isocrystal.…

Let $k(v)$ be the residue field at $v$, let $\kappa$ be a perfect field over $k(v)$, and let $x\in \mathscr{S}_K(\kappa)$. For each prime $\ell$, write $I_{\ell,x}$ for the corresp…

Integral-model conjecture. The composition

Let $X$ be a Shimura variety with special cycles $\mathcal Z(m)$ whose generating series is modular in cohomology, and let $X^*$ be its Satake–Baily–Borel compactification. Looijen…