12 problems

Let $U_m$ be the affine Markoff surface … where $m$ is an integer, and let $U_m(\mathbb{A}_{\mathbb{Z}})$ denote its integral adelic points. Ghosh–Sarnak conjecture. The number of…

Let $p$ be a prime, and let … be a Markoff surface over $\mathbb{F}_p$. Call a level $k$ generic when it contains no orbits associated with $A_4$, $S_4$, or $A_5$. Let…

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…

For each prime $p$, let $\mathcal{G}_p$ be the corresponding Markoff modulo-$p$ graph. Super Strong Approximation. The collection of Markoff modulo-$p$ graphs $(\mathcal{G}_p)_p$ f…

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 each integer $m$, let $\mathcal{U}_m$ denote the corresponding integral Markoff surface, and let $\mathbb{A}_{\mathbb{Z}}$ denote the adelic space used to formulate the integra…

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 $a,k\in\mathbb Z$, and let … be the Markoff surface over $\mathbb Z$, with $\mathcal G_{\mathcal M}$ the group generated by the three involutions, double sign changes, and coor…

Let $\mathbb{X}$ be the affine surface defined by … For a prime $p$, let $X^{*}(p)$ be the set of nonzero solutions in $\mathbb{Z}/p\mathbb{Z}$, and partition it into blocks obtain…

Let $g\in\operatorname{PSL}_{2}(\mathbb{Z})$ be hyperbolic. Let $L(g;p)$ denote the longest orbit of $g$ on $\mathbb{Y}_{-2}(\mathbb{F}_{p})$, where…

Let $H(p)$ be the permutation group induced by the action of $\operatorname{Out}(\mathbf{F}_{2})$ on $\mathbb{Y}_{-2}(\mathbb{F}_{p})$, and let … Here…

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