76 problems

McCallum–Sharifi conjecture. For all primes $p$, the pairing $\langle\cdot,\cdot\rangle_p$ is nontrivial.

Let $G_n$ denote the Gauss factorial, and call a prime $p$ 1-exceptional for $m=3$ when the corresponding exceptional congruence holds. Dummit–Ford–Kisilevsky–Sands conjecture. The…

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…

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 $\alpha$ 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…

Let $G$ be a finite group and let $\chi$ be an irreducible complex character of $G$. Let $\mathbb{Q}(\chi)$ be the field generated by the values of $\chi$, and let $c(\chi)$ be the…

Let $G\subset S_n$ be a transitive permutation group and let $Q$ be a global field with $(|G|,\operatorname{char}(Q))=1$. Let $b(G,Q)$ denote the relevant logarithmic exponent and…

Outer-derivation conjecture. Under the hypotheses of the determinant conjecture, $D$ is outer if and only if

Let $\mathbb{K}=\mathbb{Q}(\zeta)$ be the $p^k$th cyclotomic number field, where $k\in\mathbb{N}\setminus\{1\}$ and $p$ is a rational prime, and let…

Innerness conjecture. Under the hypotheses of the determinant conjecture, if $1\le e_1\le r-1$ and $e_2\ge1$, then $D$ is inner if and only if

Let $\mathbb{K}=\mathbb{Q}(\zeta)$ be the $n$th cyclotomic number field with $n=2^r p$, where $r\in\mathbb{N}$ and $p$ is an odd rational prime, and let…

Let $p$ be an odd prime, let $K=\mathbb{Q}(\zeta)$ with $O_K=\mathbb{Z}[\zeta]$, and let $A$ be the matrix defined by … for $\beta=\sum_{i=0}^{p-2}b_i\zeta^i\in K$. Let $D$ be a…

The determinant-$p$ conjecture. The matrix $A=[a_{ij}]$ is a $(p-1)\times(p-1)$ matrix with determinant $p$.

Let $q$ be a prime, and let $\kappa(q)$ denote the Euler–Kronecker constant quantity defined in the paper for the maximal real cyclotomic subfield of $\mathbb Q(\zeta_q)$. The Eule…

Let $q$ be a prime, let $h_1(q)$ be the relative class number of the cyclotomic field $\mathbb Q(\zeta_q)$ over its maximal real subfield, and define … The Kummer conjecture. As…

Let $q$ be a prime number, and let $h_1(q)$ denote the relative class number. Define … The ratio $R(q)$ is called the Kummer ratio. Kummer's conjecture. As $q$ tends to infinity, ……

For $m$ sufficiently large, let ${\mathbb Q}(\zeta_m)$ denote the $m$-th cyclotomic field and let $\gamma_{{\mathbb Q}(\zeta_m)}$ be its Euler–Kronecker constant. Ihara's growth co…

Let $K_N=\mathbb{Q}(\zeta_N)$ be the cyclotomic field of conductor $N$, where $\zeta_N$ is a primitive $N$th root of unity. A cyclotomic field is unit reducible if it has the unit-…

Let $m$ be a positive integer, let $\mathbb{Q}(\zeta_m+\zeta_m^{-1})$ be the maximal real subfield of the $m$th cyclotomic field, and let $h^+(m)$ denote its class number. Polynomi…

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$, let $X_n=2\cos(2\pi/2^{n+2})$, and let $RE_n^+=\ker({\rm N}_{n/n-1}:E_n\to E_{n-1})$. Define … The…

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$. For the norm map ${\rm N}_{n/n-1}:E_n\to E_{n-1}$, set $RE_n^+=\ker({\rm N}_{n/n-1})$, and define…