132 problems

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…

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

Let $F$ be a number field, let $\widetilde{F}$ be the compositum of all $\mathbb{Z}_{p}$-extensions of $F$, and let $K(\widetilde{F})$ be the maximal unramified abelian pro-$p$ ext…

Let $L/K$ be an abelian CM extension of number fields, with $K$ totally real, and let $G=\operatorname{Gal}(L/K)$. For finite sets of places $S,T$ of $K$, assume $S$ contains all r…

Brumer's conjecture.

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

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

Let $K$ be a nowhere-evenly-ramified number field of even degree $n$ with real signature $(r_1,r_2)$. Enumerate such fields by discriminant. Cohen–Lenstra–Martinet–Malle conjecture…

Burns's conjecture. One has $\Theta(\phi)\in\mathbb{Z}[G]$, moreover $\Theta(\phi)\in\operatorname{Ann}_{\mathbb{Z}[G]}(Cl_S(K))$, and if $S'$ satisfies…

Let $K$ be a number field, let $n=[K:\mathbb{Q}]$, and let $\epsilon>0$. Write $Cl(K)[l]$ for the subgroup of elements of the class group $Cl(K)$ annihilated by $l$, and let $D_K$…

Let $R$ be a complete local Cohen–Macaulay ring satisfying Serre's condition $(R_2)$, meaning that $R_{\mathfrak p}$ is regular for every prime $\mathfrak p$ of height at most two.…

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 $l$ 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 $K/\mathbb{Q}$ 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…

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

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 $x\to\infty$,