8 problems

Let $V\subset\mathbb{P}^N$ be an irreducible projective variety, and let $C(V)$ and, where relevant, $\gamma(V)$ be the constants in the Projective Dimension Growth Conjecture. Uni…

Generalized sumset pattern conjecture. There exists an infinite set $B\subset\mathbb N$ such that, for every $i=1,\ldots,k$,

Let $\mathfrak{f}_{c,d}$ denote the free class-$c$ nilpotent Lie rings on $d$ generators defined over $\mathbb{Z}$. For $c,d\in\mathbb{N}$, let…

Let $K$ be a number field, $S$ a set of finite places of $K$, and $g\geq 2$ an integer. Let $\mathcal{F}$ be a one-parameter family of complex abelian varieties $A$ of dimension…

Bogomolov–Tschinkel's uniform common-torsion conjecture. There exists a uniform constant $B$ such that

Let $f$ be a complex polynomial automorphism of Hénon type such that $|\operatorname{Jac}(f)|\ne 1$. For an algebraic curve $C$, write $\deg(C)$ for its degree, and write $\deg(f)$…

Let $F$ be a number field and let $g>0$. For each prime $\ell$, let $\mathscr{A}(F,g,\ell)$ be the set of $F$-isomorphism classes of $g$-dimensional abelian varieties $A/F$ such th…

Uniform Version. Let $g>0$ and $n>0$. Then there exists a bound $N=N(g,n)>0$ such that $\mathscr{A}(K,g,\ell)=\varnothing$ for any $K\in\mathscr{F}(\mathbb{Q},n)$ and any prime…