38 problems

Let $K$ be a finitely generated field, let $V$ be a divisorial set of discrete valuations of $K$, and let $G$ be a connected reductive $K$-group. For each $v otin V$, let $Kv$ deno…

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…

Graham–Lagarias–Mallows–Wilks–Yan conjecture. Every sufficiently large integer satisfying the congruence obstructions for an integral circle packing occurs as a curvature in that p…

Let a generalized circle packing be an integral circle packing, meaning that all its circle curvatures are integers, and let its modular restrictions be the congruence conditions m…

Let a primitive integral Apollonian packing be a packing whose circle curvatures are integers with greatest common divisor $1$. For a fixed packing, let the modular restrictions be…

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…

Let $F$ be an integer polynomial in $n$ variables, and order its integer points by a height function. For each prime number $p$, consider the density of $p$-adic points…

Exponential local-global principle. The sequence $u$ has no zero term $u_{n}$ if and only if there is an integer $m\geqslant 2$ such that $\gcd(b,m)=1$ and every term $u_{n}$ does…

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

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

Graham–Lagarias–Mallows–Wilks–Yan density conjectures. The congruence obstruction is essentially the only obstruction, so the set of curvatures in a primitive integral Apollonian c…

Fabrykowski–Subbarao conjecture. There should exist a nonnegative integer $k$ such that

Fabrykowski–Subbarao conjecture. If $F$ is a multiplicative function that is a local power map modulo infinitely many primes $\ell$, then $F$ is a global power map; equivalently, t…

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…

CTPS local-global conjecture. The local-global principle always holds for $G$ over $F$.

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,…

Global-to-local finiteness conjecture. The map

Exceptional-image conjecture. For primes $\ell \geq 13$, all exceptional images $G_E(\ell)$ are known, and

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 $(R_i)_{i\in I}$ be a family of subrings of $\mathbb{Q}$ finitely generated over $\mathbb{Z}$, and write … For each subring $A$ of $\mathbb{Q}$, let $\mathcal{P}(A)$ denote the…

Bartels's conjecture. The Hasse norm principle should hold for every $k$-adequate extension.

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 $i>0$. Let…