304 problems

Let $k$ be a field, let $p$ be a prime, and let $n\geq 3$. For cohomology classes … consider the $n$-fold Massey product…

Let $k$ be a number field and $G$ a transitive permutation group of degree $n$. Write $\mathcal{F}_{\disc,k}(G;X)$ for the set of $G$-extensions $L/k$ with $|\disc(L/k)|\leq X$, wh…

Greenberg's generalized conjecture. The module $X(\widetilde{k})$ is pseudo-null as a $\mathbb{Z}_p\llbracket\operatorname{Gal}(\widetilde{k}/k)\rrbracket$-module.

Let $K$ be a number field and let $K^{\text{cyc}}$ be the cyclotomic $\mathbf{Z}_p$-extension of $K$. The Iwasawa $\mu$-invariant is the coefficient governing the exponential term…

Denef–Lipshitz conjecture. For every number field $L$, the extension

For a field $k$, say that it is $C_2$ if, for every form $F\in k[X_1,\ldots,X_n]$ of degree $d$ with $n>d^2$, the equation … has a nontrivial solution in $k^n$. Artin's conjecture.…

Let $K$ be a number field, let $p$ be a prime, and let ${\mathcal T}_{S_p}$ denote the relevant tame ramification module for the maximal extension of $K$ unramified outside the pla…

Let $K/F$ be an abelian extension of number fields, let $S$ be a finite set of places containing all infinite places of $F$ and all places ramified in $K/F$, and suppose that a dis…

A universal ternary classical quadratic form over a totally real number field is a positive definite classical quadratic form in three variables that represents every totally posit…

Number-field linear independence conjecture. The numbers

The p-adic Brauer–Siegel conjecture. There exists a constant ${\mathcal C}_p({\mathcal K})$ such that

Let $L/k$ be a finite algebraic extension of number fields, and let $\zeta_L(s)$ and $\zeta_k(s)$ denote their Dedekind zeta functions. Dedekind's conjecture. The quotient … is ent…

Assume, in addition to the Stark setup, that $F$ is totally real, $K$ is a CM-field, $v$ is a finite place above a rational prime $p$, and $S$ contains all places of $F$ above $p$.…

Ozaki's non-freeness conjecture. The group $\mathcal{G}(k^\mathrm{c}_\infty)$ is not a non-abelian free pro-$p$-group.

Let $G$ be a finite group of order $n$, and let $b\mathcal{M}'(G,n,X)d$ denote the subset of $b\mathcal{M}(G,n,X)d$ consisting of number fields $K$ that admit a tower … in which ea…

Let $k$ be a number field, let $G$ be a transitive permutation group of degree $n$, and let $N\normal G$. Write $q:G\to G/N$ for the quotient map and … for the induced pushforward.…

Let $K$ be a number field, and let $A,B,C$ be non-zero elements of $\mathcal{O}_K$. Let $\Omega$ be the subgroup of roots of unity inside $\mathcal{O}_K^{\times}$. Suppose … for an…

Let $F$ be a number field of degree $d=[F:\mathbb{Q}]$, and let $E/F$ be an elliptic curve. Write $E(F)[\operatorname{tors}]$ for the subgroup of $F$-rational points of finite orde…

Let $\mathbb{B}_n$ be the number field introduced above, and let $h_n$ denote its class number for $n\in\mathbb{Z}_{\geq 1}$. Weber's conjecture. For every…

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…

Conjectural tail estimates. For every $\epsilon>0$,

Let $K$ be a number field. A solution $(x,y,z)\in K^3$ of the Fermat equation … is trivial if $xyz=0$ and non-trivial otherwise. The asymptotic Fermat conjecture. If…

Kraus's conjecture. There are no elliptic curves over $K$ with full $2$-torsion and conductor $d513$.

Leopoldt's conjecture. The $\mathbb{Z}$-rank of $\mathcal{O}_M^{\times}$ equals the $\mathbb{Z}_p$-rank of the topological closure of $D(U)$ in…

Let $K$ be a number field, and let … denote the Galois group of the maximal unramified extension of $K$. Fontaine–Mazur's unramified conjecture. Every continuous Galois representat…