55 problems

Let $p$ be any prime, let $\alpha,l\in\mathbb{N}$, and let $n,r\in\mathbb{Z}$ with $n\ge 2p^{\alpha}-1$. The theorem gives a lower bound for the $p$-adic valuation of the relevant…

Let $l$ be a positive integer and let ${\boldsymbol{s}}\in\mathbb N^l$. Define $J({\boldsymbol{s}}|2)$ to be the set of nonnegative integers $n$ for which the numerator of the mult…

Let $p\ge7$ be a prime with $p\equiv3\pmod4$, and write $n=r(p-1)$ with $r\ge1$. Odd-multiple valuation conjecture. … Equivalently, among multiples of $p-1$, … for $p\equiv3\pmod4$…

Let $p$ be an odd prime and define … with $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. The mod-4 valuation conjecture. For $1\le n\le p-2$, … and for $4\le n\le p-2$, … These formulas are s…

Let $p$) be a fixed prime, let $d$ be a positive integer, and let … be a $\mathbb{Z}_p^d$-extension tower in which $K_n/K$ corresponds to the subgroup $(p^n\mathbb{Z})^d$. Write…

Finiteness conjecture for logarithmic class groups. The set

Let $r\in\N$, let $p$ be a prime, and let $\bfs\in\N^d$ satisfy $p>|\bfs|+1$, where $|\bfs|$ is the weight and $Z_{p^r}(\bfs)$ is the corresponding finite Mordell--Tornheim sum. Fi…

D--Kakde's Matrix Coefficient Conjecture. If $\det(M)=0$, then there exist nonzero vectors $w,v\in\mathbf Q^n$ such that

Gross--Kuz'min conjecture.

For each integer $n\geq 1$, let $\left\{{\mathcal{R}}_p^{m,n}(T)\right\}_{m\geq 3}$ be the series of recurrence polynomials from the stated theorem. Eventual stability conjecture.…

Let $e$ be the order of $\mathfrak p$ in $G_{\mathfrak f}$, and suppose that $\mathfrak p^e=(\pi)$ with $\pi$ totally positive and $\pi\equiv1\pmod{\mathfrak f}$. Let $\mathcal D$…

Let $u_{\mathfrak p}$ be the Brumer–Stark element attached to the data $(S,T,H,\mathfrak P)$, and let $u_1,u_2,u_3\in F_{\mathfrak p}^*\otimes\mathbb Z[G]$ be the three formulas de…

Let $F_p^*$ be the relevant local multiplicative group and let $\widehat{F}_p^*=F_p^*\widehat\otimes\mathbf{Z}_p$. Define $v_p(\mathfrak b)^{\mathrm{an}}\in\widehat{F}_p^*$ from th…

Let $F$ be a totally real field, let $H/F$ be a finite abelian CM extension, and let $\mathfrak p$ split completely in $H$. For a prime ideal $\mathfrak b$ prime to the prescribed…

The $b7$-Leopoldt conjecture. The map $\log_{\Sigma,p}$ is injective. This is a $b7$-component form of Leopoldt's conjecture and is used to control the non-vanishing of the relevan…

Recursive uniformizer and stability conjecture. For each prime $p\geq 3$ and integer $n\geq 1$, there exists a series of polynomials $\{\mathcal R_p^{m,n}(T)\}_{m\geq 2}$ such that…

Let $M/K$ be a rank-one motive with coefficient ring $R$, let $T$ be its $p$-adic realization, and put $r=r(M/K,R)$. Assume that the $\mathcal{R}_p$-module…

Let $p\equiv 5\pmod{8}$ be a prime, and let $(a)_k=a(a+1)\cdots(a+k-1)$ denote the rising factorial. The p-adic supercongruence conjecture. … This is presented as a stronger versio…

Let $p$) be a prime, and let $\zeta(\mathfrak{s})$ and $\zeta_p(\mathfrak{s})$ denote the real-valued and $p$-adic multiple zeta values associated with an admissible index…

Let $p$ be an odd prime, let $r$ be a positive integer, and let $\delta\in\{1,2\}$. Guillera–Zudilin-type conjecture. … The paper states that this stronger modulus cannot be proved…

Let $A$ be an abelian variety, let $F/k$ be a finite Galois extension with group $G$, let $J\leq G$ with $E=F^J$, and let $p$ be a prime. Let $A^t$ be the dual abelian variety, let…

3-adic valuation pattern conjecture. There exists a 3-adic integer $\alpha$ such that, for odd $n\geq 3$, the quantity $\xi_3(L_n-1)$ depends only on $\xi_3(n-\alpha)$ and the last…

Postnikov's 5-adic valuation conjecture. For $n\geq 4$,

Generalized 2-adic valuation conjecture. There exists a 2-adic integer $\alpha_k$ and a nonnegative integer $c_k$ such that

Postnikov's 2-adic valuation conjecture. There exists such a 2-adic integer $\alpha$ satisfying