17 problems

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 $k$ be a number field, let $G$ be a transitive permutation group, and let $p$ be a prime such that $p\nmid \lvert G\rvert$. For a real number $X>1$, let $E_k(G,X)$ be the set o…

Let $\Gamma$ be a finite group and let $p$ be a prime with $p\nmid |\Gamma|$. Let … Let $K_0$ be a number field with $u$ infinite places containing the $p^r$th roots of unity but n…

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…

Cohen–Lenstra–Martinet conjecture. If $P$ is finite, then

Let $t$ be a positive integer, let $p$ be an odd prime in the degree-$2$ case, and let $p\geq 5$ be prime in the degree-$3$ case. Write $(p)_\infty=\prod_{k\geq1}(1-p^{-k})$ and…

Cohen-Martinet conjecture.

Let $K$ be a cyclic extension of $\mathbb{Q}$ of odd prime degree $q$, let $p$ be a prime not dividing $q$, and let $h_K$ denote the class number of $K$. Let $\omega$ be the multip…

Limit-measure conjecture. The measures $mu_g$ converge to a measure $mu$ on $\mathcal{E}$ such that the expected number of surjections from a $mu$-random element to every element o…

Fix a prime $p\geq 5$ and an integer $m>1$ not divisible by $p$. Let $K_0$ be a $p$-rational quadratic imaginary field, and let $\mathcal F_m(K_0)$ be the family of cyclic extensio…

Let $d=-4$. For primes $p,q$ with $pq\to\text{?}$, write $\mathrm{Cl}(dpq)$ for the relevant narrow class group, and let $\mathrm{rk}_4$ and $\mathrm{rk}_8$ denote its 4-rank and 8…

Enhanced Cohen–Lenstra conjecture. The $\mathbf{n}$-Cohen–Lenstra distribution $\mathrm{CL}_{\mathbf{n},S}$ accurately models the joint distribution of

The refined Soundararajan conjecture. As $h\to\infty$ through odd values,

Let $\ell$ be a prime, let $\mathcal{G}$ denote the set of finite abelian $\ell$-groups, and suppose that $A\in\mathcal{G}$ has $\ell$-rank $r$. Let $K_0$ be a number field contain…

Delaunay's moment conjecture. For every positive integer $\ell$, partition $\lambda=1^{m_1}2^{m_2}\cdots \ell^{m_\ell}$, and nonnegative integer $u$, the average over…

Delaunay–Jouhet moment conjecture. For every positive integer $\ell$ and partition $\lambda=1^{m_1}2^{m_2}\cdots \ell^{m_\ell}$, the average over fundamental negative discriminants…

Modified class-group distribution conjecture. A given finite $p$-group $H$ occurs as the Sylow $p$-subgroup of a relative class group $\operatorname{Cl}(K/K_0)$ for…