10 problems

Unique compatible block decomposition conjecture. The valuation data of $O/K$ admits a unique compatible block decomposition.

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…

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…

Let $\bar p \in \bar C_0$ be an odd double point, and let $T$ be one connected component of $\bar C_0\setminus\{\bar p\}$ containing $2N+1$ marked points. Odd double point local ge…

Octad picture stable-reduction conjecture. The stable reduction type, including whether or not it is hyperelliptic, of any smooth plane quartic over any non-archimedean local field…

Multiplicity-independence conjecture. The multiplicity of the building blocks does not affect the stable reduction, although it does affect arithmetic invariants, as in the case of…

Cayley octad valuation-data reduction conjecture. The valuation data of $O$ determines the stable reduction type of $C/K$.

Cayley octad combinatorial reduction conjecture. Combinatorial data about the configuration of these eight points fully determines the stable reduction type of the curve, analogous…

Let $(X,L)$ be a normal projective variety over a number field $K$, and consider integral projective models $(\mathcal{X},\mathcal{L})$ over rings of integers $\mathcal{O}_{K'}$ of…