95 problems

For a prime $p$ and a positive integer $\alpha$, define $s(p,\alpha)$ to be the least positive integer $k$ such that, for any integers $a_1,\ldots,a_k$ not congruent to $0$ modulo…

Let $p$ be a prime, let $x\in\mathbb{R}$, and let $||\cdot||$ denote the distance to the nearest integer. The $p$-adic Littlewood conjecture. For every prime $p$ and every real num…

Let $q>1$ be a power of $3$, and define … The quotient $S_q/q^2$ is an integer. Sun–Tauraso's conjecture. … Equivalently, the central binomial coefficient sum is congruent to $q^2$…

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…

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…

Reserved density conjecture.

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 $l$ be a positive integer and let ${\boldsymbol{s}}=(s_1,\dots,s_l)$ be a sequence of positive integers. For $n\geq 0$, write the multiple harmonic sum as…

Let $k$ be a totally real number field, let ${\mathfrak{f}}\neq\mathcal{O}_k$ be a proper integral ideal, let $K=k({\mathfrak{f}})$, let $G=\operatorname{Gal}(K/k)$, and let $S$ be…

Let $k$ be a totally real number field, let ${\mathfrak{f}}\neq\mathcal{O}_k$ be a proper integral ideal, let $K=k({\mathfrak{f}})$ be the associated ray-class field, and write…

Giampietro--Darmon conjecture. This value should still be algebraic, with its norm prescribed by Equation. The conjecture extends the result beyond the class-number-one setting, wh…

Let $\mathbf G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n$ and let $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. Explicit small-prime valuation conjecture. For $p=3$, every $n\ge1$ satisf…

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 an odd prime and let $r$ be a positive integer. Let $A,B\subset (\mathbb{Z}/p^r\mathbb{Z})^2$ be such that their densities satisfy $\delta_A=\delta_B\gg p^{-1}$. For…

Finiteness conjecture for logarithmic class groups. The set

Let $p$ be a prime, let $n\geq 2$, and let $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. For an element $\alpha\in \bbL_p$, write $[C_{\frac{1}{p-1}}(\alpha)]$ for the coeff…

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…

Rezk's conjecture. The Coleman norm and Ando's algebraic criteria are closely related.

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 $u_2=\sum_{\sigma\in G}u_2(\sigma)\otimes[\sigma^{-1}]\in F_{\mathfrak p}^*\otimes\mathbb Z[G]$ be the element obtained from the Eisenstein cocycle and the cap-product construc…

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 $\mathcal D$…