645 problems

Let $X$ be an elliptic curve, and for each finite place $v$ of good reduction let $a_X(v)$ be defined by $|X(k_v)|=p_v^{f_v}+1-a_X(v)$, where $p_v^{f_v}=|k_v|$. Sato–Tate conjectur…

Let $G\leq S_n$ be a transitive permutation group. Define the index function by … Let $N(G,X)$ count degree $n$ number fields with Galois group $G$ and absolute discriminant bounde…

Iwasawa's conjecture. For every cyclotomic $\mathbb{Z}_p$-extension of a number field, the $\mu$-invariant is $0$.

Let $K$ be a totally real number field. A universal ternary classical quadratic form or universal ternary classical quadratic lattice over $K$ represents every totally positive alg…

Linnik's conjecture. There is a constant $c_d>0$ such that

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, and call an extension $L_1/L_2$ of number fields integrally Diophantine when the ring of integers of $L_2$ is Diophantine in the ring of integers of…

Let $\mathbb{K}$ be a number field, let $\zeta_{\mathbb{K}}(s)$ be its Dedekind zeta function, and call a zero nontrivial if it lies in the critical strip $0<\operatorname{Re}s<1$.…

Let $P_j\in\mathbb{Z}[u]$ for $j=1,2,\ldots,d-1$, and consider the parametric Thue equation … The trivial solutions are $(1,0)$, $(0,1)$, and $(P_j,1)$ for $j\in\{1,2,\ldots,d-1\}$…

Scholz's degree-comparison conjecture.

Equidistribution conjecture. The equidistribution result known for cubic, quartic, and quintic number fields should also hold for number fields of degree $n>5$.

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 $L/K$ be a finite Galois extension of number fields with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramifie…

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…

Let $N_d(X)$ denote the number of number fields of degree $d$ having discriminant with absolute value at most $X$. Let $q(n,k)$ denote the number of partitions of $n$ into at most…

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $p$ be a prime. Consider cyclic extensions $L/\mathbb{Q}$ with Galois group isomorphic to $\mathbb{Z}/p\mathbb{Z}$. David–Fe…

Let $K$ be a number field and let $E$ be an elliptic curve over $K$. Uniform boundedness conjecture. The order of the rational torsion subgroup $E(K)_\mathrm{tor}$ is bounded by a…

Number-field linear independence conjecture. The numbers

Let $F$ be an imaginary quadratic field, let $F_\infty$ be its unique $Z_p^2$-extension, and let $\Lambda(\Omega)$ denote the Iwasawa algebra associated with…

Gross's conjecture. For every prime $p$, there exists a non-solvable number field $K/{\mathbf Q}$ ramified only at $p$ and possibly at $\infty$.

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

Stark conjecture. The Rubin–Stark element $\eta_{E/K,S(E)}^V$ should belong to the corresponding integral lattice, namely

Let $G$ be an even group and let $H\leq G$ have index $2$. For a quadratic field $K$, let $f(K)$ count the unramified extensions $L/K$ whose Galois group over $\mathbb{Q}$ is isomo…

Let $F \subset \mathbb{C}$ be a number field not contained in $\mathbb{R}$, and let $\mathcal{M}$ be the set of manifolds with invariant trace fields contained in $F$, … Let…

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