19 problems

Let $\mathcal{P}$ be an integral circle packing, let $\mathcal{K}\subset\mathbb{Z}$ be its set of curvatures, and let $\mathcal{K}_a$ be the set of integers passing all local obstr…

Let $F$ be a global field of characteristic $p\geq 0$, let $l\neq p$ be a prime, and for each place $v$ let $F_v$ be the completion of $F$ at $v$. Let $X$ be the scheme under consi…

Consider the purely exponential Diophantine equation … where the variables $x_{ij}$ are positive integers. Skolem's conjecture. If this equation has no solution, then there exists…

Global localization conjecture. For $j>0$, there is an isomorphism

Haag–Kertzer–Rickards–Stange conjecture. The sporadic set $S({\mathcal P})$ is finite.

Let $\boldsymbol u=\langle u_n\rangle_{n=-\infty}^{\infty}$ be a simple rational LRBS taking values in $\mathbb{Z}[\frac{1}{b}]$ for some non-zero integer $b$. A sequence has no ze…

The local-global multiplicity conjecture. If $t\asymp X$ is admissible, then

Let $\Gamma(2)'$ be the commutator subgroup of the level-2 principal congruence subgroup $\Gamma(2)\subset\operatorname{SL}_2(\mathbb{Z})$, and let an integer $t$ be admissible if,…

Principal homogeneous space Hasse principle. If $Y(F_\nu)\neq\varnothing$ for every $\nu\in\Omega_F$, then $Y(F)\neq\varnothing$.

Projective homogeneous space Hasse principle. If $Y(F_\nu)\neq\varnothing$ for every $\nu\in\Omega_F$, then $Y(F)\neq\varnothing$.

Let $K$ be a global field, let $X/K$ be a smooth projective geometrically integral variety of dimension $d$, let $n$ be prime to the characteristic of $K$, and let $i>0$. Set…

Let $\mathcal A$ be a finite alphabet, let $\mathfrak D_{\mathcal A}$ be its denominator set, let $\mathfrak U_{\mathcal A}$ be the set of admissible integers, and let…

Local-global conjecture for $K$-Apollonian packings. There exists an $M$ dividing $24$ such that all sufficiently large integers whose residues modulo $M$ lie in $S_M$ occur as red…

Graham–Lagarias–Mallows–Wilks–Yan conjecture. All sufficiently large integers with residues in $S$ occur as curvatures in $\mathcal P$.

Let $F$ be a number field and let $G$ be a finite group on which $\Omega_F$ acts trivially. Define … and set $K\mathcal R(O_FG)=\operatorname{Im}(\Psi)$. For each finite place $v$…

Let $f:\mathbb{Q}\longrightarrow\mathbb{Q}$. For each prime $p$, define … where $\mathbb{Z}_{(p)}^\times$ is the group of elements of $\mathbb{Q}$ whose numerator and denominator a…

Pollio–Rapinchuk's conjecture. If every extension $P$ of $k$ contained in $L_1\cap L_2$ satisfies the norm principle, then the pair $L_1,L_2$ satisfies the multinorm principle. It…

Hensley's conjecture. The denominators contain every sufficiently large natural number if and only if the Hausdorff dimension exceeds one half:

Let $K$ be a number field, let $X/K$ be smooth and projective, let $S$ be a finite set of places as in the source, and let $m,s$ be integers. Global-to-local injectivity conjecture…