76 problems

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…

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

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, ……

Assume $N=1$, and let $X^+$ denote the plus eigenspace of the cyclotomic Iwasawa module $X$ under complex conjugation. Kummer–Vandiver conjecture. … This is the cyclotomic Kummer–V…

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…

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 $l$ be a prime, let $A=Cl(\mathbb Z[\mu_l])_l$, and let $A^{[i]}$ be the eigenspace corresponding to the $i$-th power of the Teichmüller character…

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…

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 $p$.