176 problems

For every integer $d\ge 2$ and every number field $K$, there exists a constant $\epsilon_{K,d}>0$ such that…

For every smooth projective curve $C/\mathbb{Q}$ with genus $g(C)\geq 2$ and every integer $n\geq 1$, let…

Let $K$ be a number field, let $f\in K(x)$ have degree at least $2$, and let $\alpha\in\mathbb{P}^{1}(K)$. Define the backward orbit…

In a standard uniform formulation: for every choice of complexity bounds for a subvariety $X$ and every integer $r\ge 0$, there is a constant $C$ such that, for every semiabelian v…

For each integer $d\geq 5$, determine the least integer $B(d)$ such that the following holds: for every homogeneous form $F\in\mathbb{Q}[x_1,\ldots,x_n]$ of degree $d$, let…

Let $K$ be a number field, let $S$ be a finite set of finite places of $K$, and let $X=\mathbb{P}^{1}\smallsetminus\{0,1,\infty\}$. Kim's conjecture asserts that the motivic Chabau…

Let $\pi:\mathcal{A}\to S$ be an abelian scheme over an algebraically closed field of characteristic $0$, and let $X\subseteq\mathcal{A}$ be an irreducible closed subvariety. If th…

Let $A$ be an abelian variety over a number field, let $V\subseteq A$ be an irreducible algebraic curve not contained in any proper algebraic subgroup of $A$, and define…

Let $\mathcal{Y}$ be an irreducible subvariety of ${\bf G}_m^n$ defined over ${\bf C}$. For algebraic subgroups $H\subset {\bf G}_m^n$ satisfying … consider the union of the inters…

Vojta's Main Conjecture. Given $\epsilon > 0$, there exists a Zariski-closed subset $Z=Z_\epsilon eq X$ such that

Lang's conjecture. (a) Weak form: The set $X(k)$ of $k$-rational points is not Zariski dense in $X$. (b) Strong form: There exists a proper Zariski-closed subset $Z\subset X$ such…

The Lang–Vojta conjecture. Under these hypotheses, every set of $(D,S)$-integral points in $X$ is Zariski degenerate. These problems are central in Diophantine geometry and, for th…

Let $E/\mathbb{Q}$ be an elliptic curve with $j$-invariant $j_E$ and minimal discriminant $\Delta_E$. Let $\hat h$ denote the canonical height and $h$ the relevant height on ration…

Pink's conjecture. The intersection

Let $X$ be a smooth projective variety defined over a number field $k$. Bombieri–Lang conjecture. There exists a proper Zariski closed subset $\Xi\subsetneqq X$ such that, for ever…

Kim's conjecture. For $n\gg 0$,

Torsion Anomalous Conjecture. An irreducible subvariety $V$ of a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties.

Let $F\in\mathbb{Z}[x_1,\ldots,x_n]$ be an absolutely irreducible form of degree $d\geqslant 2$, and let $X_F\subset\mathbb{P}^{n-1}$ be the hypersurface defined by $F$. For…

Let $D$ be a diophantine subset of $\mathbb{C}\left(z\right)$, and for each $n\in\mathbb{Z}_{\geq 1}$ let $\mathbb{C}\left[z\right]_n$ be the $(n+1)$-dimensional $\mathbb{C}$-vecto…

Baragar–Bourgain–Gamburd–Sarnak conjecture. The Markoff equation $\mathbb{X}$ satisfies strong approximation at every prime $p$. Equivalently, $\operatorname{Out}(\Pi)$ acts transi…

Let $(X,\Delta)$ be a smooth proper orbifold curve over a number field $K$, where $\Delta=\sum_i(1-\frac{1}{m_i})D_i$, and call $(X,\Delta)$ Mordellic if, after every finite field…

Vojta's multiplier-ideal conjecture. For every real $\varepsilon>0$ and positive integer $r$, there is a proper Zariski-closed subset $Z\subsetneq\mathcal X$, depending only on…

Strong Lang–Vojta conjecture. The pair $(X,D)$ is of log general type if and only if there exists a proper closed subset

Let $K$ be a number field or a one-dimensional function field of characteristic zero, and let $n\ge 3$. Let $[Z_1:\cdots:Z_n]$ be the standard homogeneous coordinates on…

Wilkie's conjecture. There exist constants $c=c(X)$ and $d=d(X)$ such that, for every $H\in\mathbb{N}$ with $H>e$,