28 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 $K$ 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…

For each prime power $q$, let $p_{q,n}$ denote the distributions on the relevant finite $\mathbf{Z}_{(S)}[\Gamma]$-modules arising from the function-field family as the parameter…

Liu's Cohen–Lenstra equidistribution conjecture. The groups $I_e\cdot e\mathrm{Cl}(K)$, as $K$ varies over totally real $A$-extensions of $\mathbb{Q}$, should be equidistributed ac…

Let $Q$ be a global function field and let $G\subset S_n$ be a transitive permutation group with $(|G|,\operatorname{char}(Q))=1$. Let $\operatorname{inv}$ be a counting invariant.…

Let the product of ramified primes be used as the counting invariant for extensions. Product-of-ramified-primes conjecture. This ordering is conjectured to be always suitable for o…

Let $\Gamma$ be a finite abelian group, let $p$ be a prime, and let $Q$ be either ${\mathbb Q}$ or ${\mathbb F}_q(t)$ with $\gcd(p|\Gamma|,q)=\gcd(p,q-1)=1$. Let…

Let $\Gamma$ be a finite abelian group. For a Galois extension $K/{\mathbb Q}$ with $\operatorname{Gal}(K/{\mathbb Q})\simeq\Gamma$, genus theory determines a special quotient of…

For an integer $n\geq1$ and a positive real number $X$, let $\mathcal N_n^+(X)$ and $\mathcal N_n^-(X)$ denote the numbers of square-free integers $0<d\leq X$ for which the class n…

Refined error-term conjecture. For every prime $p$ and every finite abelian $p$-group $G$, there exist explicit constants $B_{\pm,p}(G)$ and exponents satisfying

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 $\lambda$ be a partition, let $G_{\lambda}(p)$ denote the corresponding finite abelian $p$-group, and let $c(\lambda)$ be the associated partition parameter. Holmin et al.'s co…

Let $\lambda$ be a partition, let $G_{\lambda}(p)$ denote the corresponding finite abelian $p$-group, and let $c(\lambda)$ and $n$ be the associated partition parameters. Let…

Localized Arakelov class group conjecture. The two displayed equalities hold for every such finite Galois extension and set $P$.

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…

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…

Moment uniqueness conjecture. The model yields a unique probability measure with the property that all relevant moments equal $1$.

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…

Let $K/K_0$ range over quadratic extensions of $K_0=\mathbb{Q}(\sqrt{-3})$, and consider their relative class groups, whose class numbers are being tested for coprimality to $3$. M…