10 problems

Let $\mathfrak V$ be a subvariety whose generic point has ordinary reduction at $p$, and let $\tilde{\mathfrak y}$ be the Abel–Jacobi image of its reduction. Let…

Let $K$ be a number field and let $X/K$ be a smooth projective variety. Say that $X$ has good ordinary reduction at a prime $v$ of $K$ when it has good reduction there and the redu…

Let $K$ be a number field and let $X$ be a smooth projective variety defined over $K$. A prime of $K$ is a place of good reduction at which the reduction of $X$ is ordinary. Serre'…

Serre's ordinary reduction conjecture. After possibly passing to a finite extension of $L$, the set of primes of the resulting number field at which $A$ has ordinary reduction has…

Let $A$ be an abelian variety over a number field $F$. A prime of $F$ is ordinary for $A$ if $A$ has good ordinary reduction. Density-one conjecture. The density of ordinary primes…

Let $A$ be an abelian variety of dimension $g$ over a number field $F$. A prime $v$ of $F$ is ordinary for $A$ if $A$ has good ordinary reduction at $v$; equivalently, if $p$ is th…

Serre's ordinary-density conjecture. There is a finite extension $L/K$ such that a positive-density set of primes $v$ of $L$ gives $X_L$ a good ordinary reduction at $v$.

Serre's ordinary reduction conjecture. The reduction of $A$ is ordinary at a density-one set of primes.

Serre's positive-density ordinary reduction conjecture. There is a positive density of primes $v$ of $K$ for which $X$ has good ordinary reduction at $v$.

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$. A model $\mathcal{A}$ of $A$ over the ring of integers $\mathcal{O}_K$ of a number field $K$ satisfies Hypothesi…