32 problems

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

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 $K_a=\mathbb{Q}(\rho_a)$ be a Shanks simplest cubic field, where $\rho_a$ is a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, and let $\varepsilon,\delta\in\mathbb{Z}[\rho_a]^\times$ sati…

Let $K$ range over totally real cubic fields, and let its unit shape be the corresponding point of the space of shapes $\mathcal{S}_2$. David–Shapira's density conjecture. The set…

Let $j_l:=(\mathbb{U}_l:\langle -1,\epsilon_l,\epsilon_l-1\rangle)$ be the unit index associated with the cubic field and units above. Louboutin's weak Ennola conjecture. For every…

Let $l\geq 3$ be an integer, and let $\epsilon_l$ generate a non-Galois totally real cubic field whose minimal polynomial is … The units $\epsilon_l$ and $\epsilon_l-1$ are excepti…

Let $N_+(X)$ and $N_-(X)$ denote the numbers of cubic fields of positive and negative discriminant, respectively, with absolute discriminant at most $X$. Shintani's conjecture. The…

Cusick's conjecture. For every $\boldsymbol{\omega}$, one has $c_s(\boldsymbol{\omega})\leq\frac{2}{7}$, and the only vectors for which $c_s(\boldsymbol{\omega})$ is near…

Cusick's conjecture. There is some integral basis in the cubic field of discriminant $49$ described above whose simultaneous Diophantine constant is $\frac{2}{7}\approx0.286$, and…

Kim–Ostlund conjecture. The spiral field should give the generalization of the noble numbers for two-dimensional maps: robust two-tori should have rotation vectors associated with…

trace-minimum conjecture.

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…

Let $H(d)[3]$ denote the relevant $3$-torsion statistic of the class group attached to a negative fundamental discriminant $d$, and consider its Davenport–Heilbronn asymptotic aver…