124 problems

Let $p$ be an odd prime, let $K_p^+=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ be the maximal real subfield of the $p$th cyclotomic field, and let $h_p^+$ be its class number. The plus part…

Let $m$ be a prime, let $K_m=\mathbb{Q}(\zeta_m)$ be the cyclotomic field generated by a primitive $m$-th root of unity, let $K_m^+=\mathbb{Q}(\zeta_m+\zeta_m^{-1})$ be its maximal…

Fix an even character $\theta:(\mathbb{Z}/Np\mathbb{Z})^{\times}\rightarrow\overline{\mathbb{Q}}_p^{\times}$ and let $\omega$ be the Teichmüller character. Assume…

Let $q$ be a prime, and let $\gamma_q$ denote the Euler–Kronecker constant associated with the cyclotomic field considered in the paper. The Ihara positivity conjecture. … always.…

Let $p>3$ be prime, let $K=\mathbb Q(\zeta)$ be the $p$-th cyclotomic field, and let $u,v\in\mathbb Z\setminus\{0\}$ be coprime. The equation … according as $u+v\not\equiv0\pmod p$…

Let $p$ be a prime and let $r$ be a positive integer. Write $h_{p^r}^+$ for the class number of the maximal real subfield $\mathbb{Q}(\zeta_{p^r})^+$ of the cyclotomic field…

Let $\mathcal{F}$ be the module of functions on roots of unity satisfying the relevant domain conditions, and let $\mathcal{F}^{\rm d}\subseteq\mathcal{F}$ be the $\mathcal{R}$-mod…

Let $l$ be an odd prime, let $A$ be the $l$-torsion part of the class group of $\mathbb Z[\mu_l]$, and let $\omega$ be the Teichmüller character of $G(\mathbb Q(\mu_l)/\mathbb Q)$.…

Let $p$ be an odd prime, let $K=\mathbb Q(\zeta)$ be the $p$-th cyclotomic field, and consider a solution of the SFLT2 equation. Reduced-form conjecture. If the SFLT2 conjecture fa…

Nonvanishing determinant conjecture. $\det(N_\ell)\neq 0$ for every odd prime $\ell$.

Let ${\cal Z}(N)$ be the algebra of multiple polylogarithm values at $N$-th roots of unity, filtered by weight, and let $UC_{\bullet}(N)^{\vee}$ denote the graded dual of the unive…

Smallness conjecture for $\nu$. Thus we conjecture that $\nu$ is in most cases small before $p$.

Smallness conjecture for $c1$. We conjecture that $c1$ is in most cases small before $p$.

Let $p>5$ be prime, let $x,y,z\in\mathbb{Z}-\{0\}$ be mutually coprime and satisfy $x^p+y^p+z^p=0$ with $xyz\not\equiv 0\pmod p$, and let $t\equiv -x/y\pmod p$. Consider the explic…

Surjectivity conjecture. The natural map is surjective. This conjecture concerns whether Milnor K-theory accounts for the relevant $p$-adic Quillen K-theory under Vandiver's conjec…

Let $p$) be an odd prime, let $K={\mathbb Q}(\mu_p)$, and let $\mathcal{O}_K$ be the ring of $p$-integers in $K$. For an irregular pair $(p,r)$, let … denote the pairing constructe…

Goncharov's polynomiality conjecture. If $p$ is a prime, then $d_w(p)$ and $d_{w,m}(p)$ are polynomials in $p$.

Goncharov's isomorphism conjecture. The map $\nu_{\bullet,\bullet}^{\cal M}(\mu_N)$ is an isomorphism for either $N=1$, or $N=p$ is a prime and $w=m$.

Let $p$ be a prime, let $E_m(N)$ be the arithmetic coefficient module defined in the source, let $M^*_{(m)}$ be the dual modular complex, and let…

Let $p$ be a prime number, let $\boldsymbol{}[p]$ be the group of $p$-th roots of unity, let ${D}^{\Delta}_{\bullet}(\boldsymbol{}[p])$ be the diagonal, depth-equals-weight part of…

Let $M^*_{(m)}$ be the modular complex, let $\Gamma_1(m;N)$ be the indicated congruence subgroup, let ${\rm V}_m$ be the representation appearing in the source, and let…

Let ${D}_{\bullet\bullet}(G)$ be the bigraded dihedral Lie algebra for a finite commutative group $G$, let ${D}_\bullet(G)$ be its quotient by components with unequal weight and de…

Cyclotomic phase-locking conjecture. Cyclotomic aspects in phase-locking could play an important role in many fundamental tests of quantum mechanics related to quantum entanglement…

Let $x_1,\ldots,x_r$ be roots of unity, let $a\in\mathbb{C}\setminus\mathbb{Z}$, and let $\bfk=(k_1,\ldots,k_r)\in\mathbb{N}^r$ satisfy $(k_1,x_1)\neq(1,1)$ and…

Let $\alpha$ be a cyclotomic integer, meaning an algebraic integer expressible as a sum of roots of unity. Let $l(\alpha)$ be the smallest number of roots of unity occurring in any…