138 problems

van der Waerden's conjecture. The number of such exceptional polynomials satisfies

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

Poonen–Rains conjecture. As $E$ varies over all elliptic curves over $F$ ordered by height,

Let $E$ be an elliptic curve over $\mathbb{Q}(t)$, let $E(\mathbb{A}^1)$ denote the elliptic curve over the function field $\mathbb{Q}(t)$, and let $E(t)$ denote its specialization…

Let $K$ be a number field, and let elliptic curves over $K$ be ordered by height. For an elliptic curve $E/K$, write $r$ for its Mordell–Weil rank. Goldfeld–Katz–Sarnak conjecture.…

Brumer–McGuinness conjecture. The number of elliptic curves with bounded discriminant satisfies

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 $C$ be a smooth projective curve over a number field $k$. Let $X_k$ be the set of conjugacy classes of $\operatorname{ST}(\operatorname{Jac}(C)_k)$, and let $\{p_i\}_{i\geq 1}$…

Let $\mathbb{F}_q(t)$ be the rational function field over the finite field $\mathbb{F}_q$, and consider cubic extensions $K/\mathbb{F}_q(t)$ counted by discriminant. Secondary-term…

Let $E$ range over elliptic curves over $\mathbb{Q}$, and let $\operatorname{rank} E(\mathbb{Q})$ denote the Mordell–Weil rank of $E$ over $\mathbb{Q}$. Park–Poonen–Voight–Wood's c…

Fix a prime $p$, and let $\mathcal{S}_p$ be the set of elliptic curves $E/\mathbb{Q}$ such that $\Sha(E/\mathbb{Q})[p^\infty]\neq 0$. Let $\bar{\mathfrak{d}}$ denote upper density…

Let $K$ be a number field and let $\mathscr{X}/\mathcal{O}_K$ be a model for a hyperbolic curve. Let $\mathcal{L}$ be an allowable family of extensions $L/K$. Write…

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 $n$ be a positive integer, and let $E$ range over elliptic curves, ordered by naive height. The $n$-Selmer group of $E$ is denoted by $\operatorname{Sel}_n(E)$. Bhargava–Kane–L…

Roberts' conjecture. There are positive constants $c_1$ and $c_2$ such that

Ohno–Nakagawa reflection conjecture. A reflection theorem holds for pairs of $n$-ary quadratic forms for every odd $n$.

Malle-type conjecture for algebraic tori. For every $n\geq 1$ and every finite subgroup $1\neq H\leq\operatorname{GL}_n(\mathbb{Z})$, there are positive integers $a(H),b(H)$ and a…

Let $K$ be a number field, and order elliptic curves over $K$ by height. Rank distribution conjecture. Half of all elliptic curves over $K$ have Mordell–Weil rank zero, while the r…

Let $n$ be any positive integer. For elliptic curves $E$ over $\mathbb{Q}$ ordered by height, let $\operatorname{Sel}_n(E/\mathbb{Q})$ denote the $n$-Selmer group, and let…

Let $E/\mathbb{Q}$ be a non-CM elliptic curve. For each positive integer $k$, let $E[k]$ be the $k$-torsion subgroup and let $\mathbb{Q}(E[k])$ be the field obtained by adjoining t…

Density conjecture. Except for a set of $t\in\mathbf P^1(\mathbf Q)$ of density $0$, the rank of ${\mathscr E}_t(\mathbf Q)$ is either

Parity conjecture. One expects

For the family of elliptic curves … let the average be taken over the family ordered by the relevant height, and let the associated $2$-isogeny Selmer group be the Selmer group of…

Folklore discriminant-counting conjecture. The number of degree-$n$ number fields with discriminant bounded in absolute value by $X$ is asymptotic to a constant times $X$.

For an integer $n\geq 2$, a positive integer $\Delta\geq 1$, and $\varepsilon>0$, let $K$ range over number fields of degree $n$ satisfying $|\Delta_K|=\Delta$, where $\Delta_K$ is…