27 problems

Let $D$ be a $p$-divisible group, with codimension $c$ and dimension $d$. Its isomorphism number $n_D$ is the smallest non-negative integer such that, for every $p$-divisible group…

Let $G$ be a smooth affine group scheme over $\mathbb{Z}_p$, and let $\mu:\mathbb{G}_m\to G_{\mathcal{O}}$ be a $1$-bounded cocharacter defined over the ring of integers…

Very-good integral Hodge embedding conjecture. Any good integral Hodge embedding is also very good.

Unconditional local Shimura conjecture. The natural transformation

Let $\mathcal{N}_{s,n}$ be the moduli space of quadruples defining the Rapoport–Zink space for $\operatorname{GU}(s,n-s)$ over a ramified prime, with $n\geq 3$ and $s\leq n$, and s…

Let $G$ be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ and let $\mu:{\mathbb{G}}_m\to G$ be 1-bounded. Let $\operatorname{BT}_n^{G,\mu,?}$ be the quotient 2-st…

Let $G$ be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ equipped with a 1-bounded homomorphism $\mu:{\mathbb{G}}_m\to G$. Let $\hat{\mathscr R}_n$ be the stack…

Let $G$) be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ equipped with a 1-bounded cocharacter $\mu:{\mathbb{G}}_m\to G$. Let $\operatorname{BT}_n^{G,\mu}$ be t…

Let $\mathfrak{S}=W(k)[[t]]$ and let $\mathcal{E}$ be an Eisenstein polynomial. Let $\mathrm{BT}(R)$ be the category of $p$-divisible groups over the relevant regular local ring, a…

Invariant-subvariety conjecture. The special fiber of $\mathfrak M_{\mathcal R,\Omega}$ contains a nonempty $Z(\mathbb A^{\infty,p})\times G(\mathbb A^{\infty,p}\otimes L^+)$-invar…

Let $Y$ be the object used to define the formally smooth deformation space $\mathbf{Def}_{\mathrm{sus}}(Y)$, and let $\mathfrak{a}_Y$ be its ambient Lie algebra. Let…

Let $K$ be a characteristic zero local field with ring of integers $\mathcal{O}_K$, residue field $k$, and algebraic closure $\bar{k}$. Let $\mathscr{G}$ be a $p$-divisible group o…

Let $\mathit{k}$ be the residue field. Consider a Rapoport–Zink space and the corresponding affine Deligne–Lusztig set, with their natural geometric structures; write the perfectio…

Let $G$ be a reductive group over $\mathbb{Q}_p$, let $\mu:\mathbb{G}_{m\overline{\mathbb{Q}}_p}\to G_{\overline{\mathbb{Q}}_p}$ be a cocharacter, and let $\nu$ be a Harder–Narasim…

Let $G$ be a reductive group over $\mathbb{Q}_p$, let $\mu:\mathbb{G}_{m\overline{\mathbb{Q}}_p}\to G_{\overline{\mathbb{Q}}_p}$ be a cocharacter, and let…

The arbitrary-Newton-polygon conjecture. For an arbitrary Newton polygon $\xi$, let $w\in{}^J W$ be a generic specialization of $w_\xi$. Then there exists a Newton polygon $\zeta$…

Versal-family pullback conjecture. Does there exist a variety $Y$ and a height-$2$, dimension-$1$ $p$-divisible group $\mathscr{H}$ on $Y$, everywhere versally deformed, together w…

Let $\mathbb X_0$ be a $p$-divisible group over a field $k$ of characteristic $p$, and let $\nu$ be its Newton polygon. Let $\nu'\in {\mathcal N}(G)_{{\mathbb Z}}$ with…

Let $(N,b\sigma)$ be an isocrystal. Let $J_b(F)$ be the group acting on the associated affine Deligne-Lusztig variety, let $M=gM_0$ be a Dieudonné lattice, and let…

Let $G=GL_n$ and let $[b]$ be a $sigma$-conjugacy class as in the paper. Let $f_g:J_b(F)\rightarrow X_*(T)_{{\rm dom}}$ be the invariant defining the stratification, and let the ge…

Let $F$ be the relevant base field, let $O_D$ be the specified maximal order, and fix an $r$-special $p$-divisible group $({\mathbb {X}},\iota_{{\mathbb {X}}})$ over an algebraic c…

Let $\Bbbk$ be an algebraically closed field of characteristic $p>0$, and let $D$ be a $p$-divisible group over $\Bbbk$. Its isomorphism number $n_D$ is the smallest nonnegative in…

EL-chart bijection conjecture. The construction of $\tilde\mu$ induces a bijection

Let $\widehat{\mathcal{M}}$ be the formal moduli space of $p$-divisible groups with the additional structures under consideration, and let $\triangle$ be the target of the morphism…

Let $k$ be a perfect field of characteristic $p$, let $W(k)$ be its ring of Witt vectors, let $K_0=W(k)[1/p]$, and let $K$ be a finite totally ramified extension of $K_0$ with ring…