132 problems

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

Let $K/\mathbb{Q}$ be a number field of degree $s$ with discriminant $D_K$. For an integer $\ell\geq 1$, write $h_K[\ell]$ for the order of the $\ell$-torsion subgroup of the class…

Let $F$ be a number field, let $H/F$ be a finite abelian extension with Galois group $G$, and let $S$ and $T$ be disjoint finite sets of places of $F$ such that $S$ contains the ar…

Finiteness conjecture. There are only finitely many primes $\ell$ for which the $\ell$-group of logarithmic classes of $K$ is nontrivial.

Generalized projective Euler characteristic conjecture. There exists an integer $Q(d)$ such that, whenever $G$ is finite and ${\rm \gcd}(\#G,Q(d))=1$, one has

Let $p$ be a prime, let $K$ be a number field, and let $\widetilde{K}$ be the compositum of all $\mathbb{Z}_p$-extensions of $K$. Let $X$ be the Galois group of the maximal abelian…

Cohen–Lenstra conjecture. The proportion of discriminants $D\in S_X^{\pm}$ for which the Sylow $p$-subgroup of the class group of $\mathbb{Q}(\sqrt D)$ is isomorphic to $B$ satisfi…

Generalized Gross conjecture. The coinvariant group $X'(k^\mathrm{c}_\infty)_{\operatorname{Gal}(k^\mathrm{c}_\infty/k)}$ is finite.

Weak form of Greenberg's conjecture. If $X(k^\mathrm{c}_\infty)$ is nontrivial, then it has a nontrivial finite $bZ_p\llbracket\operatorname{Gal}(k^\mathrm{c}_\infty/k)\rrbracket$-…

Greenberg's conjecture. The module $X(k^\mathrm{c}_\infty)$ is finite.

Let $S$) be a finite set of primes not containing any divisor of $2|\Gamma|$, let $\Gamma_\infty\subset\Gamma$ have order $1$ or $2$, and let $\mathcal{F}(\Gamma_\infty,X)$ be the…

Let $p$ be a prime, let $k'$ be the quadratic field and $\varepsilon'$ a unit of $k'$ occurring in the algorithm, and let $h_{k'}$ denote its class number. Let $\eta_1$ and…

Let $K$ be a totally real number field of degree $n$. For an odd prime $p$ unramified in $K/\mathbb{Q}$ and a place $\mathfrak{p}$ above $p$, let $\tau^{\mathfrak p}$ be the map ap…

Let $Q={\mathbb Q}$ or $\mathbb F_q(t)$, and let $\mu_Q$ be the group of roots of unity contained in $Q$. Let $\Gamma$ be the finite group and $\Gamma_\infty$ the cyclic subgroup g…

Stevenhagen's conjecture. As $x\to\infty$,

Malle's conjecture. The average size of $\operatorname{Cl}_2(K)$ over totally real cubic fields $K$ is $5/4$, and its second moment is $15/8$; over complex cubic fields $K$, the av…

Let $n>1$ be an integer, let $F$ be a number field, and let $d$ be a positive integer. For a number field $L$, write $Cl(L)[n]$ for the subgroup of the ideal class group of $L$ ann…

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 $K=\mathbb{Q}(\sqrt{D})$, where $D=pq$ with $p\equiv 5\pmod 8$ and $q\equiv 1\pmod 8$, and suppose … Assume also … Let $X_{\infty}$ be the Iwasawa module of the cyclotomic…

Let $p$ be an odd prime, let $K$ be a CM field, and let $K^+$ be its maximal totally real subfield. Let $S^+$ be a set of primes of $K^+$ above $p$, each of which splits in $K$ as…

Class-number infinitude conjecture. There are infinitely many $r\in\mathbb{Z}^+$ such that $p$ does not divide the class number of $K_r$.

Let $k$ be a number field, let $G$ be a transitive permutation group, and let $p$ be a prime such that $p\nmid \lvert G\rvert$. For a real number $X>1$, let $E_k(G,X)$ be the set o…

Finiteness conjecture for logarithmic class groups. The set

Let $\chi = \chi_0 \chi_p$, with $\chi_0 \ne 1$, be an even irreducible rational character and let $K=K_\chi$. For every irreducible $p$-adic character $\varphi$ dividing $\chi$, s…

Gross--Kuz'min conjecture.