60 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…

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 $G$ 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 $N_d(X)$ 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…

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

Malle's prediction. If $\operatorname{Aut}_{\overline{K}}(C)\cong D_4$, then

Let $n\in\mathbb{Z}_{\geq 0}$ and suppose that $\operatorname{char}(\mathbb{F}_q)>3$. Let $\mathcal{N}^{r}_{T}(\mathbb{F}_q(t),B)$ count the number of $\mathbb{F}_q$-isomorphism cl…

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 $\mathbb{N}^*$ denote the positive integers. For squarefree $s$, write … and for a function $f\colon\mathbb{N}^*\to\mathbb{C}$ define its average over the squarefree integers c…

Arithmetic kei coloring average-order conjecture. Let $\mathscr{K}\in\mathscr{K}\operatorname{ei}$ be finite. Then there exists $a_{\mathscr{K}}>0$ such that

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 $\{p_i\}_{i\geq 1}$…

Average-size conjecture. The average size of the $n$-Selmer groups is

Secure-torsor counting conjecture. If $F$ is a number field, there exists $C>0$ such that

Let $p$ be an odd prime. Let $\mathfrak{T}_p$ denote the set of elliptic curves $E_{/\mathbb{Q}}$ such that the $p$-primary part of the Tate–Shafarevich group…

Uniformity conjecture. For every $M>0$,

Dion–Ray conjecture. The set $\mathcal{U}^p$ has density

Fix a prime $p$. For elliptic curves $E/\mathbb{Q}$ ordered by height, consider the proportion of curves for which $p\mid\#\Sha(E/\mathbb{Q})$. Delaunay's proportion conjecture. Th…

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. A prime $p$ is a local torsion prime when $E$ has good reduction at $p$ and $E(\mathbb{Q}_p)[p]\neq 0$. Davi…

Let $n=2m$, let $S$ be a finite set of primes, let $T$ be a finite set of primes (or an infinite set when $n>2$) disjoint from $S$, and let $X_{S,T}$ be the set of $S_n$-number fie…

Let $p$ be a prime and let $n=2m$. Write $\operatorname{Sel}_2^{\mathbb{Q}}(F)$ for the rational part of the 2-Selmer group of an $S_n$-number field $F$, and let $c(n,p)$ denote th…

Let $n$ be a positive integer, and let $\Sigma=(\Sigma_2,\Sigma_3,\Sigma_5,\ldots)$ be an acceptable collection of local specifications, where each $\Sigma_p$ is a set of…

Let $\mathscr{E}$ be the set of isomorphism classes of elliptic curves over ${\mathbb Q}$, ordered by height, and let $\mathscr{E}'$ be the subset consisting of rank-zero elliptic…

Nagao's conjecture. The rank $r$ of $\mathcal{E}(\mathbb{Q}(T))$ satisfies

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 $n$ 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…