16 problems

Let $\mathbb{K}$ be a number field, and let $\tilde\Gamma\subseteq\overline{\mathbb{Q}}^\times$ be a subgroup of finite rank. Write $\tilde\Gamma^{\mathrm{div}}$ for its divisible…

Function-field Vojta conjecture. Let $\mathcal{A}$ be an ample line bundle on $\mathcal{X}$ and let $r$ be a positive integer. For every $\epsilon>0$, there is a proper Zariski-clo…

For an individual rational function $f\in\operatorname{Rat}_d(k)$, let $h_{\operatorname{crit},\mathbb{Q}}$ be the critical height on the moduli space $\mathrm{M}_d$, and let…

Let $X/K$ be a smooth projective variety over a number field, let $H$ be a Weil height associated to an ample divisor, and define the arithmetic order $\mathsf{A}(X)$ using the ite…

Let $X$ be a projective variety over a number field $K$, let $f:X\dashrightarrow X$ be a dominant rational self-map, and let $V$ be an irreducible subvariety of dimension $k$. Supp…

For a stacky curve $\mathfrak{X}=\mathfrak{X}(\mathbb{P}^1:({\mathbf a},{\mathbf m}))$, let $\gamma(\mathfrak{X})$ be the infimum of the real numbers $\delta$ for which the modifie…

Let $\mathfrak{X}$ be a “nice” algebraic stack defined over a number field $K$, and let $\mathcal{E}$ be a “nice” vector bundle on $\mathfrak{X}$. Let…

Let $X$ be a quasi-projective variety, let $\Phi\colon X\dashrightarrow X$ be a rational self-map, and let $f\colon X\dashrightarrow \mathbb{P}^1$ be a non-constant rational functi…

Let $X$ be a quasi-projective variety, let $\Phi\colon X\dashrightarrow X$ be a rational self-map, and let $f\colon X\dashrightarrow \mathbb{P}^1$ be a non-constant rational functi…

Vojta's function-field conjecture. For every $\varepsilon>0$ there exist a constant $C$ and a proper closed subvariety $Z$ such that, for all…

For an integer $d\geq 3$, let $f\in\mathbb{Q}[x]$ be a polynomial of degree $d$ with nonzero discriminant, and let $h(f)$ denote the maximum of the logarithmic heights of its coeff…

Let $g_1,\dots,g_n\in\overline{\mathbb{Q}}^\times$ and let $\Gamma=\langle g_1,\dots,g_n\rangle^{\mathrm{div}}$ be a finite-rank divisible subgroup of…

Vojta's conjecture. The stated inequality holds for all such closed points outside $Z$.

Let $S/K$ be a variety, let $A/S$ be a principally polarised abelian scheme, and let $\sigma\in A(S)$ be a section of infinite order. For an integer $d>0$ and $\epsilon\geq0$, defi…

Vojta–Hall–Lang conjecture. There exist constants $C_1=C_1(d)$ and $C_2=C_2(d)$ such that

Let ${\mathcal C}\to B$ be a family of higher-genus curves, let $b\in B({\mathbb Q})$, and let ${\mathcal C}_b$ be a smooth fiber. Let $H_B$ be a non-logarithmic height on $B$, and…