10 problems

Baragar–Bourgain–Gamburd–Sarnak conjecture. The Markoff equation $\mathbb{X}$ satisfies strong approximation at every prime $p$. Equivalently, $\operatorname{Out}(\Pi)$ acts transi…

Let $p$ be a prime and let $\mathbf{M}_k$ be the Markoff surface … Let $\mathcal{E}_k$ denote the union of the exceptional orbits in $\mathbf{M}_k(\mathbb{F}_p)$, and let…

Let $\mathcal{M}$ be the set of Markoff triples over $\mathbb{Z}$, let $\mathcal{M}_p$ be the set of their reductions modulo a prime $p$, and let $\mathcal{G}^\times_p$ be the Mark…

Let $\mathcal{X}$ be the affine surface in $\mathbb{A}^3$ defined by … and let $\Gamma$ be the group of affine integral morphisms generated by coordinate permutations and the Vieta…

For an integer $m$, let $U_m$ be the affine Markoff surface … For $m=0$, let $U_0^*(\mathbb{Z}/p\mathbb{Z})=U_0(\mathbb{Z}/p\mathbb{Z})\setminus\{(0,0,0)\}$. Let $\Gamma$ be the gr…

Let $k$ be a number field, let $P_1(t),\cdots, P_n(t)$ be pairwise distinct irreducible polynomials in $k[t]$. Set $k_i=k[t]/(P_i(t))$, let $a_i$ be the class of $t$ in $k_i$, let…

Optimal strong approximation conjecture. For every $\epsilon>0$, as $q\to\infty$, there exists a set $Y\subset G_q$ with

Let $N,m,\lambda_1,\dots,\lambda_4\in\mathbb{Z}$ satisfy … Four-square strong approximation conjecture. If $N\gg m^{4+\varepsilon}$, then there exists…

Strong Approximation Conjecture. For any prime $p$, $\mathbb X(\mathbb Z/p\mathbb Z)$ consists of exactly two $\Gamma$-orbits, namely

Harari–Wittenberg's conjecture. The set