194 problems

Katz's Sato–Tate conjecture. For every $f\in\mathcal{C}([0,\pi])$,

Let $\mathcal{M}_n(P_1)$ be the set of Stoimenow matchings avoiding $P_1$, let $\mathcal{P}_n{\bf (3+1)}$ be the corresponding class of $(3+1)$-avoiding Fishburn posets, and let…

Equidistribution conjecture. The equidistribution result known for cubic, quartic, and quintic number fields should also hold for number fields of degree $n>5$.

Let $0\le s\le k$ be an integer. Let $f:\mathbb{P}^k\to\mathbb{P}^k$ be a holomorphic endomorphism of degree $d\geq 2$, and let $T$ be its Green $(1,1)$-current. For an analytic su…

Equidistribution conjecture. For every integer $N>1$ and every $a\in\mathbb{Z}/N\mathbb{Z}$,

Let $X$ be the quotient under consideration, let $h(t)$ denote the horocycle flow, let $p\in X$ be non-periodic, and let $\mu_X$ be the invariant measure. For a sequence $a_n$, the…

Let $d$ be a locally admissible squarefree positive integer, and let … be the graph of the Gauß orthogonal complement, where $\mathcal{S}^2=\mathrm{SO}_3(\mathbb{Z})\backslash S^2$…

Let $\Gamma\subset PSL(2,\mathbb{R})$ be such that $X=\Gamma\backslash PSL(2,\mathbb{R})$ is not compact, and let $(h_t)$ be the horocycle flow acting on $X$. Margulis–Shah conject…

Let $D_i<0$ be fundamental discriminants, let $\operatorname{Cl}(D_i)$ be the class group of $\mathbb{Q}(\sqrt{D_i})$, let $\sigma_i\in\operatorname{Cl}(D_i)$, and let $q_i$ be the…

Gouvêa's distribution conjecture. The set $X_T$ is equidistributed on $[0,1]$ as $T$ tends to infinity.

For odd integers $k$, consider primitive quartic Dirichlet characters $\chi$ of odd conductor $q$, and let $\tau(\chi)$ denote their Gauss sums. Odd-moment equidistribution conject…

Let $S_n$ be the set of permutations of $\\{1,2,\ldots,n\\}$, and let $\operatorname{des}_2$, $\operatorname{cyc}$, $\operatorname{pex}$, $\operatorname{des}$, and…

Let $f$ be a holomorphic cusp form, and consider its zeros in shrinking balls around infinity, in the regime where the height parameter satisfies $H\gg\sqrt{K\log k}$. Ghosh–Sarnak…

Let $\mathcal{A}_n$ be the set of ascent sequences of length $n$, and let $\mathsf{asc}$, $\mathsf{rep}$, $\mathsf{zero}$, and $\mathsf{max}$ denote the corresponding Euler–Stirlin…

All-point equidistribution conjecture. The effective equidistribution result may hold for all points on any nilmanifolds.

For $i=1,2$, let $g_i=\sum_{n=1}^{\infty}b_i(n)q^n$ be primitive eigenforms of weight $2k_i$ and level $2N_i$, and define … Let $g_1,g_2$ be distinct non-CM primitive forms of weig…

Let $K$ be a number field, let $X$ be a projective variety over $K$, and let $\phi:X\rightarrow X$ be polarized by an ample line bundle $\mathcal{L}$, so that…

Equidistribution conjecture. For every generic $H$, one has

Let $K$ be a global function field, let $M_K$ be the set of places of $K$, and let $\widehat h$ be the canonical height associated to the Drinfeld module $\phi$. For…

Let $K$ be a global function field, let $\boldsymbol{\phi}$ be a Drinfeld module of rank $d$ over $K$, let $M_K$ be the set of places of $K$, and let $\boldsymbol{\phi}_Q$ denote t…

Fix $\rho > 0$. Let $G=\mathbf{G}(\mathbb{R})$ be an $\mathbb{R}$-split real algebraic group, let $\Gamma < G$ be an arithmetic lattice, and let $H$ be a maximal $\mathbb{R}$-split…

Let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of generic characteristic, let $g\geq 1$, and let $\{x_k\}\subset\mathbb{G}_a^g(K^{\operatorname{sep}})$ be a strict sequence…

Let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of generic characteristic acting diagonally on $\mathbb{G}_a^g$, and let $\{x_k\}$ be a strict sequence of torsion points in…

Dynamical Siegel conjecture. For any nonpreperiodic point $\beta\in\mathbb{P}^1_{\mathfrak{o}_K}(\overline{K})$, there are at most finitely many preperiodic points of $\varphi$ in…

Let $m\ge2$ be an integer, and let $f(X)\in\mathbb Z[X]$ represent infinitely many primes. A congruence class modulo $m$ is allowable if every integer $r$ in it satisfies…