18 problems

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

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…

Let $G$ be a reductive algebraic group over a finitely generated field $K$, and let $V$ be a divisorial set of discrete valuations of $K$. A $K$-form of $G$ is a reductive algebrai…

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

Regulated local good-reduction conjecture.

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

Scholl's conjecture. The property that $[E_{\mathfrak p}]$ has good reduction is independent of the prime $\ell$.

Let $G$ be a reductive affine algebraic group over a finitely generated field $k$, and let $V$ be a divisorial set of discrete valuations of $k$. A $k$-form of $G$ is a reductive a…

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…

Finiteness conjecture for forms with good reduction. For every reductive algebraic $K$-group $G$, there are only finitely many $K$-isomorphism classes of $K$-forms of $G$ having go…

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…

Let $K=k(C)$ be the function field of a smooth affine geometrically integral curve over a field $k$, and let $V_0$ be the set of discrete valuations associated with the closed poin…

Let $K$ be a finitely generated field, and let $V$ be a divisorial set of places. For a given absolutely almost simple algebraic $K$-group $G$, a $K$-form of $G$ is an algebraic…

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…