11 problems

Let $k$ be any complete non-Archimedean field, and let $X$ be a projective $k$-variety. A morphism $f\colon X\to X$ is polarized if there exists an ample line bundle $L\to X$ and a…

Regulated independence-of-$\ell$ conjecture. The inclusion

Regulated local good-reduction conjecture.

Integral-part conjecture. For any prime $\ell$ not lying below $\mathfrak p$,

Let $\mathscr{E}$ be the set of isomorphism classes of elliptic curves over ${\mathbb Q}$, ordered by height, and let $\mathscr{E}'$ be the subset consisting of rank-zero elliptic…

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $F$ be a finite extension of $W[1/p]$. Let $X_F$ be a K3 surface over $F$ such that…

Let $A$ be an abelian variety of dimension $g$ over a number field $F$, assume that $A$ has good reduction everywhere, and let $P_{A,\infty}(\eta)$ and $\mathfrak a_\eta$ be the ar…

Let $r\geq 2$ and $n\geq 1$. For a number field $K$ and a finite set $S$ of finite places of $K$, write $\mathcal{O}_{K,S}$ for the ring of $S$-integers. A smooth hypersurface of d…

Let $K=k(C)$ be the function field of a smooth affine geometrically integral curve over a field $k$, and let $V$ be the set of discrete valuations associated with the closed points…

Finiteness conjecture. The set of $K$-isomorphism classes of Fano threefolds over $K$ of Picard number $1$ with good reduction outside of $S$ is finite.

Shafarevich conjecture for complete intersections. The set of $K$-linear isomorphism classes of smooth complete intersections of type $T$ over $K$ with good reduction outside $S$ i…