135 problems

Let $X$ be a smooth projective variety over $\mathbb{C}_p$, and let $X^{\mathrm{an}}$ denote its analytification. Kodaira-dimension conjecture. If there is a $p$-adic analytic map…

Let $D=0+1+\lambda+\infty$ on $\mathbb{P}^1$, and consider logarithmic graded semistable Higgs bundles on $(\mathbb{P}^1,D)$ whose underlying graded vector bundle is isomorphic to…

Let $X$ be a projective smooth curve over $\mathbf C=\mathbf C_p$, the complete algebraic closure of $\mathbf Q_p$. A semistable Higgs bundle on $X$ is a Higgs bundle of degree $0$…

Let $k$ be a number field and let $X$ be a smooth projective variety over $k$. The cotangent bundle $\Omega^1_X$ is nef when it is a nef vector bundle. Nef-cotangent conjecture. If…

Let $k$ be the residue field and let $G$ and $(b,)$ be the data defining the integral moduli $v$-sheaf $\mathcal{M}^{\mathop{\mathrm{int}}\nolimits}_{G,b,\mu}$. A formal scheme is…

Let $\tau,\omega\in\mathcal{H}_p$ be RM points, with $\tau\notin\Gamma\cdot\omega$, and let $J_p(\tau,\omega)\in\mathbb{C}_p^\times$ be the value of the associated rigid meromorphi…

Let $X_p$ be the $p$-adic symmetric space attached to an orthogonal group, and let $J_{\mathscr{D}}$ be a rigid meromorphic cocycle that can be evaluated at a suitable special poin…

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 $p\geq 5$, and let $\overline{\rho}$ be a residual Galois representation that is regular in the sense specified in the paper. For each…

Let $E$ and $E'$ be $p$-adic fields, and let $Y/E'$ and $X/E$ be anabelomorphic, geometrically connected, smooth hyperbolic curves. Here, anabelomorphic means that their associated…

Let $G$ be a connected, quasisplit reductive group over $4p$, let $4mu$ be a dominant cocharacter, and let $b\in \mathbf{B}(G,\mu)$. Let $M_b\subset M_S\subset G$ be standard ratio…

Carayol's conjecture. In each setting, the construction realizes simultaneously the Jacquet–Langlands and Langlands correspondences: a tensor product…

Let $K$ be a $p$-adically closed field and let $A$ be an infinite definable subset of $K^n$ that is open in its closure. Let $(B_k)_{k\leq q}$ be a finite collection of closed defi…

Let $V$ be a smooth projective geometrically integral variety over a $p$-adic field $k$. Let $A_0(V)$ be the degree-zero subgroup of the Chow group of zero-cycles, and let … be the…

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 ${\mathbb C}_p$ be the completion of the algebraic closure of ${\mathbb Q}_p$, with $v$ the extension of the $p$-adic valuation, and let … be its ring of integers. Let…

Stable supersingular-component conjecture. There is a semistable covering of $X_0(p^3)$ defined over $R_p$ such that, for each such $A$, one connected component of the supersingula…

CM residue-disk placement conjecture. The $j$-invariants of all such curves $E$ lie in $2(p+1)/i$ residue disks inside the corresponding Atkin–Lehner circle, with $(p+1)/i$ disks f…

Moduli-theoretic component conjecture. In general, there should be components having the same moduli-theoretic properties as the components in the semistable covering of $X_0(125)$…

Supersingular component-type conjecture. In general, for each supersingular elliptic curve defined over $\mathbb{F}_p$, there should be components that look like $E_1$, $E_2$, the…

CM placement conjecture at 13. If

CM placement conjecture at 7. If

CM placement conjecture at 5. If

Let $\mathbf F$ be a finite field of characteristic $p$, let $\overline{\rho}: \mathrm{Gal}(\overline{\mathbf Q}/\mathbf Q) \longrightarrow \mathrm{GL}_2(\mathbf F)$ be a modular G…

Monodromy conjecture. The filtration $M_{\bullet}$ is the monodromy filtration. This conjecture was known in the equicharacteristic case by Deligne and in a surface case by Rapopor…