161 problems

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 $(\mathsf{G},\mathsf{X})$ 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 $k(v)$ 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

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

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…

Assume the divisor conjecture and let $pi:{rm Sh}{K1}(G,X)E$ be the generic-fiber $T{\mathcal G}(\mathbb Fp)$-torsor,…

Assume that $({\mathcal G},\{\mu\})$ is strictly convex, that the divisor conjecture holds, and that an $O_E$-integral model ${\mathcal S}_{K_0}$ of ${\rm Sh}_{K_0}(G,X)_E$ with it…

Let $F$ be the number field in the setup, let $\Pi$ be a $\mathfrak d$-REASDC representation of $\operatorname{GL}_{2m}(\mathbb A_F)$, let $E$ be a strong coefficient field of…

Let $\overline{\operatorname{Sh}}$ be the Siegel threefold or a unitary Shimura variety, and let $\mathfrak{m}\subseteq\mathbb{T}$ be generic and non-Eisenstein. Let…

Let $(G,X)$ be a Shimura datum of Hodge type. Let $\mathbb{T}$ be the spherical Hecke algebra and let $\mathfrak{m}\subseteq\mathbb{T}$ be a maximal ideal. For an algebraic represe…

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

Let $(G,X)$ be a Shimura datum with associated $G(\mathbb{C})$-conjugacy class of minuscule cocharacters $[\mu^{-1}]$ and reflex field $E_0$. Fix a prime $p$, let $E$ be the comple…

Let $\mathbf{I}$ be a finite index set, let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}$ separate factors over $\mathscr{A}_{g_{\mathbf I},\mathbb{F}}$, and for each $i\in\mathbf I$…

Let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}$ be a locally closed immersion with image in the ordinary locus. A subvariety is quasi-weakly special when its ordinary locus is conta…

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

The coherent cohomological dimension conjecture. One conjectures that

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 $(\pi,\pi')$ be a coherent pair linked by the chain of diagrams constructed in the paper to an elementary pair $(\sigma,\sigma')$. Coherent-pair cohomology conjecture. This cha…

Assume the indefinite Shimura-variety setup of the source. Let $\mathcal{S}_0^\circ$ and $\mathcal{S}_0^\bullet$ be the relevant Shimura varieties, let $\Omega_\lambda$ be the loca…

In the indefinite case, let $\mathcal{S}_0^\circ$ be the relevant Shimura variety, let $\widetilde{\mathcal{S}}_0^\circ$ be its special fiber, let $\mathrm{S}'_0$ parameterize the…

Let $X_\Gamma^\Sigma$ be a toroidal compactification of an orthogonal type Shimura variety, or of a unitary type Shimura variety of signature $(n+1,1)$. The special cycles of codim…