304 problems

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…

Bilu–Luca conjecture. There exists a real number $c>0$ such that, for every sufficiently large integer $N$, at least $cN$ of the number fields…

Let $K$ be an $S_n$-field for $n\geq 3$ and signature $(r_1,r_2)$. Let $\mathcal S$ be a finite set of local conditions, and let $|\mathcal S|$ denote the product of their local de…

Let $M$ be a hyperbolic $3$-manifold with invariant trace field $k=k(M)$, and let $\overline{k}$ denote its complex conjugate. The Chern–Simons invariant $CS(M)$ lies in…

Let $E$ be an elliptic curve over a number field. Its essential dimension is denoted by $\operatorname{ed}E$. Infinite essential-dimension conjecture for elliptic curves. Then … Th…

Strong Torsion Conjecture. If $n\geq n(g,d)$, then there are no dimension-$g$ abelian varieties $A$ defined over $K$, with $[K:\mathbb Q]\leq d$, that have a $K$-rational torsion p…

BSD regulator-quotient square conjecture. Then

Fix an integer $d \geq 3$. For a totally real number field $K$ of degree $d$, let $\operatorname{disc}(K)$ denote its discriminant, and let an ideal class mean a class of fractiona…

Let $E$ be an elliptic curve over a number field $K$. Let $w(E/K)\in\{\pm 1\}$ be the root number, defined as the conjectural sign in the functional equation for $L(E/K,s)$ under…

Let $E/\mathbb Q$ be the number field and $\bar G$ its relevant Galois group, let $E_0=\{x\in E:\operatorname{Tr}_{E/\mathbb Q}(x)=0\}$, and let $\mu_\infty$ and $\mu_p$ be the arc…

For a number field $k$, write $\mathcal{D}_k$ for its discriminant. Malle's polynomial bound conjecture. There exists an absolute constant $B$ such that, for sufficiently large $x$…

Let $S$ be a finite set of places of a number field $k$ containing all archimedean places, and let $L_S=(L_v)_{v\in S}$ be an unconditional $S$-tuple of separable quadratic algebra…

Fix a number field $F$. For each positive integer $N$, let $r_F(N)$ be the maximum of $\operatorname{Rank}_{\mathbb{Z}}E(F)$ over elliptic curves $E/F$ whose conductor…

Let $K$ be a number field and let $X/K$ be a smooth projective variety. Say that $X$ has Hodge–Witt reduction at a prime of $K$ when its smooth reduction at that prime is Hodge–Wit…

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…

Let $\Lambda$ be a finitely generated subgroup of $\overline{\mathbb{Q}}^\times$ containing $\zeta_{m-1}$, with $m\ge 2$. Say that algebraic numbers are multiplicatively dependent…

Let $K$ be a number field, let $M_K$ be its set of places, and define the height $\operatorname{h}$ and conductor $\operatorname{cond}$ as in the preceding number-field abc formula…

The abc-conjecture for a number field. For every $\kappa>1$ and every $x\in K$ distinct from $0$ and $1$,

Refined Malle conjecture for ${\rm Heis}_4$. The group ${\rm Heis}_4$ fails to satisfy independence of local events, and

Let $K$ be a number field, let $n$ be a positive integer, let $\mathfrak{R}$ be a nonzero ideal of the ring of integers of $K$, and let $G_K$ be the absolute Galois group of $K$. F…

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

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 $S$) 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 $G$ 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…

Finite generation conjecture. For every $n\in\mathbb{Z}$ and every $i\leq 2n+1$, the group