64 problems

Let $\mathcal{R}=\mathbb{C}[x,y]$, and let $N^0(F)$ denote the Newton polygon of $F$ with the origin as reference point. Let $a,b\in\mathbb{Z}_{>0}$ be relatively prime with…

Let $\Delta$ be a lattice polytope of degree denominator $D$, and let ${\rm GNP}(\Delta,p)$ and ${\rm HP}(\Delta)$ denote its generic Newton polygon and Hodge polygon, respectively…

Let $f$ be a Laurent polynomial with coefficients in $\overline{\mathbb{Q}}$, and let $\triangle$ be its Newton polytope. For each prime $p$, fix an embedding…

Let $f$ be a non-constant monic polynomial in $Q[x]$. A global permutation polynomial (GPP) over $Q$ is a polynomial $P(x)\inQ[x]$ whose reduction modulo $p$ is a permutation of…

Let $C_f(s,T)$ be the $C$-function of the $T$-adic exponential sums associated with a polynomial $f$, and let its absolute $T$-adic Newton polygon mean the $T^a$-adic Newton polygo…

Let $g',g”\in\mathbb Z_{>0}$, and let $\xi'$ and $\xi”$ be symmetric Newton polygons appearing on $\mathcal T^0_{g'}\otimes\mathbb F_p$ and $\mathcal T^0_{g”}\otimes\mathbb F_p$, r…

Let $k((t))$ be a local function field and let an overconvergent $(F,\nabla)$-crystal over $k((t))$ be given. A crystal is potentially semistable if, after a finite extension…

Wan's conjecture. There is a Zariski dense subset ${\mathcal{U}}$ in $\mathbb{A}^d$ such that, for every $f(x)\in {\mathcal{U}}$, the limit exists and

Saturation conjecture. The terms that appear in the numerator of a Markov polynomial $M_{\rho}$ are precisely those corresponding to the integer lattice points in the Newton polygo…

Let $p$ be a prime, and let $\xi_1$ and $\xi_2$ be Newton polygons that are realized in characteristic $p$. Oort's direct-sum conjecture. Then $\xi_1 \oplus \xi_2$ is realized in c…

Let $g \geq 18$, and let $m$ be the least common multiple of the exponents of all of the Galois groups of abelian covers of $\mathbf{P}^1$ branched at three points of genus $g$. Un…

Let $p$ be the residue characteristic, and consider the Hodge polygon associated with the infinite symmetric power $L$-function of the hyper-Kloosterman family, denoted by…

The Okamoto Hamiltonians for the Painlevé equations are polynomial Hamiltonians whose Newton polygons have genus $1$; for the known cases, the areas are $2$ for…

Let $\rho=a/b$ be rational, and let the critical triangle be the set of lattice points satisfying … Consider the coefficients of the Markov polynomial associated with these lattice…

Saturation conjecture. The terms appearing in the numerator of $M_\rho$ are precisely those corresponding to the integer lattice points in $\Delta_\rho$. The conjecture is an equiv…

Let $\rho=a/b$ be a rational number, and let $\Delta_{a/b}$ be the Newton polygon of the numerator of the Markov polynomial $M_\rho(x,y,z)$, consisting of the points $i,j\geq0$ sat…

Let $\overline{\mathcal{R}}=\mathbb{C}[x^{1/\mathbf d},y]$ for a fixed positive integer $\mathbf d$, and let $\overline{T}_{m,n,a}$ be the class defined in the paper. Suppose…

Let $Z$ be the innermost polynomial and $\zeta(F)$ the inner polynomial arising from the paper's construction. For a convex polygon, the NE vertex is the uppermost vertex with larg…

Let $\mathcal{Q}$ and $R_{m,n}$ be as above, and let $\zeta(F)$ denote the inner polynomial of $F$. The inner-polynomial lattice-point conjecture. If $(a,b,m,n)\in\mathcal{Q}$ and…

The structured Newton-polygon conjecture. Suppose $(a,b,m,n)\in\mathcal{Q}$ and $F,G\in\mathcal{R}$ satisfy $[F,G]\in\mathbb{C}$, $F\in R_{m,n}$, and $G\in R_{bm/a,bn/a}$. Then…

Generalized Dumas–Eisenstein irreducibility conjecture. The polynomial

Oort's conjecture. If $\xi_i$ occurs on $\mathcal{M}_{g_i}$ for $i=1,2$, then $\xi_1\oplus\xi_2$ occurs on $\mathcal{M}_{g_1+g_2}$.

Let $\bar\rho=\begin{pmatrix}\omega_1^{a+b+1}&*\\0&\omega_1^b\end{pmatrix}:\operatorname{I}_{\mathbb{Q}_p}\to\operatorname{GL}_2(\mathbb{F})$ be a character with…

The remainder vanishing conjecture. Under these conditions,

The Newton-polygon criterion. Then $[F,G]=0$.