272 problems

André-Oort conjecture. If $V$ contains a Zariski dense set of special points, then $V$ is a Shimura variety.

Let $X$ be a complex Shimura variety, let $X^{\text{BB}}$ be its Baily--Borel compactification, and write $IH^k(X^{\text{BB}};\mathbb{C})$ for its intersection cohomology and…

Let $\operatorname{Ig}_{\mathbf{b}}(\overline{k})$ be the geometric-point set of the Igusa variety, with its natural $G(\mathbb{A}^{\infty,p})\times J_{\mathbf{b}}(\mathbb{Q}_p)$-a…

Let $G$ be the reductive group defining the affine Deligne–Lusztig variety, let $\boldsymbol{\theta}$ denote the induced action on the relevant character lattices, and let…

Assume that $L(\tfrac12,\Pi)\neq0$ for the Rankin–Selberg representation $\Pi$. Let $V_n$ and $V_{n+1}$ be hermitian spaces over $F$, totally positive definite at infinity, with…

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…

André–Pink conjecture. If $V$ has Zariski dense intersection with $O$, then $V$ is totally geodesic.

Let $S=S(G,h)$ be a Shimura variety with $G\otimes_{\mathbb{Q}}\mathbb{R}=GU(r,s)$, and let $M_{r,s}$ be the associated local model over $\operatorname{Spec}(R)$, where $R$ is the…

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 $p>0$, 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 $K^p$ be a neat compact open subgroup of $d53e({\mathbb A_f^p})$, let $K=K_pK^p$, and let $(\mathsf{G},\mathsf{X})$ be a Shimura datum satisfying…

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. A Hodge class is a class in $H^{2k}(S,\mathbb{Q})$ of Hodge type $(k,k)$, for $k=0,\ldots,n$. The Hodge con…

General modular-height conjecture. For general Shimura varieties, the formula for the modular height of the integral model should, up to some constant differences, align with certa…

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

Let $X$ be a Shimura variety of Type I or Type IV, and let $X\hookrightarrow X^\Sigma$ be a smooth toroidal compactification. Let $\overline{\mathcal Z}(m)$ denote the Zariski clos…

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…

For a local Shimura datum $(G,b,\{\mu\})$, let $\mathcal{F}=\mathcal{F}(G,\{\mu\})$ be the associated flag variety, let $\breve{\mathcal{F}}^{\mathrm{rig}}$ be its rigid analytic b…

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.

Canonical integral models conjecture. For every such Shimura datum and quasi-parahoric model, there exists a system of canonical integral models…

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

Let $k$) be an algebraic closure of $\a0\mathbb F_p$, let $W(k)$ be its ring of Witt vectors with Frobenius automorphism $\sigma$, and put $L=W(k)[1/p]$. Let $G$ be a connected red…

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…