18 problems

Let $\bar{\bar \alpha}=\left(\alpha_{0},\alpha_{1},\alpha_{2}\right)$ be any $\mathbb{Q}$-basis of a real cubic number field. Let $\mathscr S\left(\bar{\bar \alpha}\right)$ denote…

Malle's conjecture. The average size of $\operatorname{Cl}_2(K)$ over totally real cubic fields $K$ is $5/4$, and its second moment is $15/8$; over complex cubic fields $K$, the av…

Let $L$ be a non-Galois totally real cubic field with ramified Galois closure $N$, conductor $f>1$ over a real quadratic field $K$, and type $\alpha$, with $U_N=U_0$. Let…

Let $f$ be a regular conductor, let $d>0$ be a quadratic fundamental discriminant, and consider $3$-admissible pairs $(f,d)$ with $3$-class rank $\varrho=1$. Let $(L_1,\ldots,L_m)$…

Let $K=\mathbb{Q}(\sqrt{d})$ be a quadratic field with $d\equiv 2\pmod 3$ and $3$-class rank $\varrho=0$. Consider $3$-admissible non-split prime or prime-power conductors $f=q$,…

Let $K=\mathbb{Q}(\sqrt{d})$ be a quadratic field with $3$-class rank $\varrho$, and let the associated unramified nilets, singlets, and quartets be classified by their types. The…

The one-level density conjecture. The displayed asymptotic formula should hold, and the two sums on its right-hand side are absolutely convergent.

Periodicity conjecture without Preliminary Stage 2. Excluding Preliminary Stage 2 from the sin-squared algorithm will not affect periodicity for cubic vectors.

Let $p$ be a prime number such that $p \neq 3$. The quantity $\mu_p(\mathcal{F}^{+}(Y))$ denotes the average size of the $p$-torsion subgroup in the class groups of the positive-di…

Let $p\neq 3$ be a prime number, let $P=p$ or $2p$, and let … be the fundamental unit of $\mathbb{Q}(P^{1/3})$, where $\alpha,\beta,\gamma\in(1/3)\mathbb{Z}$. Shimizu–Hirakawa conj…

Let $p\neq 3$ be a prime number, let $P=p$ or $2p$, and let … be the fundamental unit of $\mathbb{Q}(P^{1/3})$, where $\alpha,\beta,\gamma\in\mathbb{Z}$. The cubic analogue of the…

Malle's class-group rank conjecture. As $K$ ranges over cyclic number fields of degree $\ell$, for every $r\in\mathbb{Z}_{\geq0}$,

Let $p$ be a prime number such that $p\equiv 1\, (\mathrm{mod}\,3)$, let $L=\mathbb{Q}(\sqrt[3]{p})$ be the corresponding pure cubic field, and let…

Let $I_{0,c}$ be the initial point set associated with the positive integer $c$, whose elements are cubic algebraic integers. Distinct cubic fields conjecture. For every…

Let $p>3$ be a prime, let $K$ range over cyclic cubic fields, and let $R_{K,p}$ denote the normalized $p$-adic regulator. Hofman-Zhang's conjecture. … The source says that this pre…

Cohen-Martinet conjecture.

Let $K$ be a real cubic field, and let $\alpha^{(0)},\alpha^{(1)},\alpha^{(2)}$ be its positive $\mathbb{Q}$-basis. Set … Let $\{\varepsilon_n\}_{n=0}^{\infty}$ be the expansion of…

Let $K$ be a real cubic field and let $r=5/2$. Write $\Delta_K$ for the discriminant of $K$ and $\Delta_{K,r}^{\mathcal{P}er}$ for the associated periodic discriminant quantity. Peri…