10 problems

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…

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…

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…

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…

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…