140 problems

Let $A$ be a simple abelian variety over $\mathbb{Q}$ of dimension $n$, and let $\mathcal{L}\subset A(\mathbb{Q})$ be a finitely generated subgroup of rank $m\le n$. Let…

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 $\mathcal{X}$ denote the completely disconnected space modelling physical space at the Planck length, and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Volovich's conjec…

Let $n\geq 3$ be an integer, let $p$ be a prime satisfying $p\equiv 1\pmod n$, and let $\Gamma_p(x)$ denote the $p$-adic Gamma function introduced by Morita. Deines–Fuselier–Long–S…

Let $phi\in\mathbb{Z}[X_1,\ldots,X_n]$ and let … be the local exponential sum, where the sum is over those $x=(x_1,\ldots,x_n)$ for which $p\mid x_j$ for every $j$. Let…

Rationality conjecture. The sum is a rational number if and only if $k=x=1$. This conjecture concerns when the $p$-adic sum of the factorial power series can be rational; the suppl…

Let $p$ be a prime, let ${{\mathbb Q}}_p$ be the field of $p$-adic numbers, and let $\Omega(|x|_p)$ denote the characteristic function of the unit disc. A function $\phi$ satisfyin…

For a positive integer $N$, let $q_{N,(N)}(z)$ denote the mirror-map series used in the paper, let $\Xi_N$ be the integer defined from the $p$-adic valuations of $H_N$, and let…

Let $X$ be a projective variety defined over the finite field $\mathbb F_q$, and suppose that its effective monoid $A^+(X)$ is finitely generated. The height zeta function associat…

Denef–Sargos conjecture. The exponent $\sigma$ in these bounds is optimal for infinitely many $p$ and $m$.

$p$-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 rationality conjecture. The series $P_{f,K}(T)$ is rational in $T$ for all $K\in\mathcal{B}_k$. Rationality is known for the classes $\mathcal{A}_k$ and, for sufficiently l…

Furstenberg's conjecture. The semigroups ${\cal S}_p$ and ${\cal S}_q$ are transverse semigroups acting on ${\mathbb Z}_r$.

For an even integer $n$, let $B_n$ denote the $n$th Bernoulli number, let $|\cdot|_p$ be the $p$-adic absolute value, let $\Psi^{\rm irr}_1$ be the set of irregular pairs, and for…

Analytic-structure conjecture. Every smooth $p$-adic Banach-Lie group admits a smoothly compatible $p$-adic analytic Lie group structure.

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 $(W_n)$ be an unbounded elliptic divisibility sequence and let $p$ be a prime. Suppose there is an exponent $e\geq 1$ such that, for each $0\leq i<e$, the limit … exists in…

Let $\mathbf{S}_0(\operatorname{SL}_2(\mathbf{Z});\mathbf{Q}_{11};11^{-r})$ be the space of $11^{-r}$-overconvergent forms of weight $0$ and level $1$, where $r$ is rational and…

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 $n$ be a positive integer, and let $Y$ and $Y+1$ be consecutive integers satisfying $up(Y)+ up(Y+1)=nup(n!)$ with the full valuation $nup(n!)$ assigned exclusively to one of t…

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…

The relativistic $p$-adic sunscreen conjecture. The set