72 problems

Let $f\in\mathbb{Z}[x_1,\ldots,x_n]$ be nonconstant, and define the normalized exponential sum … where $p$ is prime and $m\geq 1$. If $\operatorname{lct}(f)$ denotes the log-canoni…

For every prime $p>3$, prove that $\displaystyle \sum_{n=0}^{p-1}(2n+1)^3v_n\!\left(\frac52\right)^2\equiv 6p^4-\frac{143}{3}p^6\pmod{p^7}$, where…

Let $\Omega\subset\mathbb{Q}_p^d$ be an admissible bounded domain, let $D_{\Omega}^{\alpha}$ denote the Dirichlet realization of the Vladimirov–Taibleson operator, and let…

Let $\kappa_r(A,B)\in\mathbb{Q}[A,B]$ be the $r$-th additive phase constant in the asymptotic phase equation for the zeros of Jacobi polynomials, with $A=\alpha^2$ and $B=\beta^2$.…

Suppose $z_1,\ldots,z_n\in\mathbb{D}_p\subset\mathbb{C}_p$ are $\mathbb{Q}$-linearly independent, where $\mathbb{D}_p$ is the domain of the $p$-adic exponential function. Define…

Let $p$ be a prime and let $b1$ be an irrational real number. Let $\lvert q\rvert_p=p^{-v_p(q)}$ denote the $p$-adic absolute value of the integer $q$, where…

Leopoldt's conjecture. The $\mathbb{Z}$-rank of $\mathcal{O}_M^{\times}$ equals the $\mathbb{Z}_p$-rank of the topological closure of $D(U)$ in…

Shell-wise p-adic Duffin–Schaeffer conjecture. One should have

$p$-adic non-vanishing conjecture. The function $\tau ^p(M)$ is nowhere zero on $U(\mathcal{O}_p)$. The source presents this as a generalization of the root-of-unity non-vanishing…

Igusa's conjecture. There exists a constant $c$ such that for every prime $p$ and every positive integer $m$,

Let ${\mathcal W}=(W_n)_{n\ge0}$ be an elliptic divisibility sequence and let $p$ be a prime. For an exponent $N=N_p\ge1$, consider the subsequences indexed by $mp^{kN}$ for each…

Let $k\geq12$ be even, let $S_k=S_k(\Gamma_0(1),\mathbf{Q})$ be the space of level-$1$ cusp forms of weight $k$, and let $m=\dim S_k$. For a polynomial with rational coefficients,…

Let $I_{\lambda_0,f}(\lambda)$ denote the Mellin transform associated with the parameters $\lambda_0$, $f$, and $\lambda$, and let $q$ be the residue-field cardinality. Rationality…

Let $\mathcal R_{\mathbb Q}=\bigcup_N\mathcal R_{1/N}$ be the rational Robba ring over $\mathbb C_p$, completed under the Gauss norm. Rational Robba-ring admissibility conjecture.…

Let $\mathfrak{O}$ be the coefficient ring, let $A$ be an $n$-dimensional nilpotent $\mathfrak{O}$-algebra, and write $\zeta_A(t)$ and $\zeta^{\lhd}_A(t)$ for its subalgebra and id…

Let $\mathfrak{O}$ be the coefficient ring and let $A$ be a finite-dimensional $\mathfrak{O}$-algebra. Write $\zeta_A(t)$ and $\zeta^{\lhd}_A(t)$ for its subalgebra and ideal zeta…

Fix an embedding $\iota_p:\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_p$, let $A$ be an abelian variety over $\overline{\mathbb{Q}}$ of dimension $d$, and let…

Wang's transformation conjecture. If $p$ is an odd prime, then

Let $\mathcal{A}$ be a commutative algebraic group over $\mathbb{Q}$ that is either a torus or an abelian variety, with Lie algebra $\mathfrak{a}$ over $\mathbb{Q}$ and exponential…

Let $\mathcal{A}$ be a connected commutative algebraic group over $\mathbb{Q}$, let $\mathcal{L}\subset\mathcal{A}(\mathbb{Q})$ be finitely generated, and let…

Let $I$ be an open polysegment, let $P$ be a $\nabla$-module over $A^n_K(I)$ satisfying the Robba condition, and let $A$ be an exponent of $P$. A multisubset $B$ of…

Let $\overline{\alpha}\in\mathbb{R}^n$ be a vector and let $p$ be a prime. Denote by $\langle\cdot\rangle:\mathbb{R}^n\to[0,1)^n$ the $\ell^\infty$ distance to the nearest integer…

Vladimirov sub-Laplacian conjecture. The Vladimirov sub-Laplacian of order $\alpha>0$ is a hypoelliptic operator on $G$ and is invertible on the space of mean-zero functions.

Let $H:\mathbb{Z}\to\mathbb{Z}$ be a contracting, semi-basic $p$-Hydra map, and let $\widehat{\chi}_H$ be a Fourier transform of its numen. Fourier-transform dynamical criterion. (…

Let $H$ be as in the source's theorem giving the multidimensional $\mathcal{F}$-series for $\chi_H$, and suppose that $\mathbf I_d-\alpha_H(\mathbf 0)$ is invertible. Let $\psi_H$…