18 problems

Let $A$ be an abelian variety of dimension $g$, let $V\subset A$ be an irreducible subvariety of dimension $d$, and let $\Gamma\subset A$ be a subgroup of finite rank. The variety…

Let $N$ be the level parameter, let $\alpha$ be the invariant defining the modular polynomials with level structures, and let $J_1$ be the corresponding polynomial. For a polynomia…

Boyd’s conjecture. The optimal constant $c$ should equal

Discriminant-negligibility conjecture. There exists a Weil height function $h\colon S(\overline{\mathbb Q})\to\mathbb R$ such that, for every special point $p\in S$ and every…

For each $\Delta>0$, let $\mathrm{NL}_\Delta$ denote the corresponding Noether–Lefschetz divisor, and let $\|\mathrm{NL}_\Delta\|_1$ be its coefficient $1$-norm. Existing results g…

Let $X$ be an integral projective variety in projective space, let $d$ be its degree, and let $N(X;B)$ denote the number of rational points of $X$ of height at most $B$. Dimension…

Schinzel–Zassenhaus–Boyd conjecture. There is a constant $T>1$ such that if $\alpha$ is not a root of unity, then

Let $X\subset G$ be irreducible and not a translate, and let $H$ be the irreducible translate of minimal dimension containing $X$. Write $\operatorname{codim}_H X$ for the codimens…

Let $G$ be a group variety defined over a number field $k$. Let $b1$ be a point of $G$, and let $B$ be the irreducible torsion variety of minimal dimension containing $b1$. The fie…

Bounded Height Conjecture. For an irreducible variety $V$ in a semi-abelian variety, the set of maximal $V$-torsion anomalous points has bounded height.

Let $\mathcal{C}\rightarrow\mathcal{B}$ be a family of positive genus curves equipped with a map $f\in K(\mathcal{C})$ that is non-constant on fibers. Let $h_{\mathcal{C}}$ be an a…

Let $Y(1)$ denote the modular curve and let $C\subset Y(1)^2$ be an irreducible algebraic curve defined over $\overline{\mathbf Q}$ that is not special. For $p\in Y(1)^2$, let…

Asymptotic height lower-bound conjecture. For prime $l$,

Height upper-bound conjecture. For every prime $l>30$,

Let ${\mathcal C}_N$ be the family of genus-$2$ curves defined by integral binary sextics with coefficients bounded in absolute value by $N$. For a rational point $P$ on such a cur…

Let $(A,\mathcal{L})$ be a polarized abelian variety of dimension $g$ defined over a number field $\mathbb{K}$, and let $X\subset A$ be a transverse subvariety with finite stabiliz…

All varieties are defined over $\overline{\mathbb{Q}}$. Let $A$ be an abelian variety of dimension $g$, let $V\subset A$ be an irreducible subvariety of dimension $d$, and let…

Bounded-height conjecture. There exist $\varepsilon>0$ and a non-empty Zariski open subset $V^u$ of $V$ such that: