9 problems

Fix a prime $p$, let $n\in\mathbb{N}$, let $G$ be a smooth affine group scheme over $\mathbb{Z}/p^n\mathbb{Z}$, and let $\mu:\mathbb{G}_m\to G$ be a cocharacter. Let…

Smoothness conjecture. The stack $\operatorname{BT}_{n,m}^{G,\mu}$ is smooth over ${\mathbb{Z}}/p^m$.

Drinfeld's conjecture. The stack $\operatorname{BT}_{n,m}^{G,\mu}$ is algebraic, and a suitable version of contravariant Dieudonné theory yields an isomorphism

Drinfeld's algebraicity conjecture. This restriction is a quasicompact smooth algebraic stack over $\mathbb Z/p^m\mathbb Z$ with affine diagonal.

Drinfeld's gerbe isomorphism conjecture. These two gerbes are isomorphic.

Drinfeld's extension conjecture. The isomorphism between these restrictions extends to an isomorphism over the whole $\operatorname{Disp}_1^G$.

Drinfeld's conjecture. The new stack is canonically isomorphic to the reduction modulo $p$ of the stack of $1$-truncated Barsotti–Tate groups of height $d$ and dimension $d'$.

Fake elliptic curve conjecture. The local system $\mathscr{L}$ should come from a family of fake elliptic curves: there exists a map $f:C\to X^{D}$ and a family…

Maximal monodromy conjecture. The monodromy of a versal family of ordinary Barsotti–Tate groups in characteristic $p>0$ is maximal.