65 problems

Rubin–Stark conjecture. The Rubin–Stark element satisfies

Let $p$ be a prime number and let $m$ be a positive integer. For an integer $d$, consider the $m+1$ successive quadratic fields … Here, “real (or imaginary) quadratic” means that t…

Let $m$ be a prime, let $K_m=\mathbb{Q}(\zeta_m)$ be the cyclotomic field generated by a primitive $m$-th root of unity, let $K_m^+=\mathbb{Q}(\zeta_m+\zeta_m^{-1})$ be its maximal…

Let $\mathbb{B}_n$ be the number field introduced above, and let $h_n$ denote its class number for $n\in\mathbb{Z}_{\geq 1}$. Weber's conjecture. For every…

Class number formula conjecture. For $n>1$,

The Möbius-inversion class-number conjecture. One has

For odd $n$, let $\Delta=n^2+4$ and let the maximal order of discriminant $\Delta$ have class number one. Yokoi's conjecture. All such maximal orders must satisfy … This is a class…

Let $p$ be a prime and let $r$ be a positive integer. Write $h_{p^r}^+$ for the class number of the maximal real subfield $\mathbb{Q}(\zeta_{p^r})^+$ of the cyclotomic field…

Brauer–Siegel conjecture. Then

Let $m$ be an integer and let $d=m^2+1$ be prime. The associated real quadratic field is $\mathbb Q(\sqrt d)$. Chowla–Friedlander conjecture. If $m>26$, then the class number of…

Let $p$ be a positive integer, and let $h(n)$ denote the class number associated with the discriminant or class-number problem in question. Chowla's conjecture. … for every $p>13$.…

Let $d>0$ range over fundamental discriminants, let $h(d)$ denote the narrow class number of the real quadratic field of discriminant $d$, and let $\varepsilon_d$ be its fundamenta…

Class-number asymptotic conjecture. Under the conditions of Theorem 1, as $x\to\infty$ one has

Let $S$ be a finite set of places of a number field $k$ containing all archimedean places, and let $L_S=(L_v)_{v\in S}$ be an unconditional $S$-tuple of separable quadratic algebra…

Generalized Brauer–Siegel conjecture. Then

Stark's conjecture. For every fixed bound on the class number, only finitely many CM-fields have class number at most that bound.

Let $\Delta=n^2-4$ be squarefree, with $n\ge 21$, and write $h_{n^2-4}$ for the class number of the corresponding quadratic order. Mollin's conjecture. … The supplied status eviden…

Let $n$ be an odd squarefree positive integer, and let $h^*(-p)=h(-p)$ for $p\geq7$ and $h^*(-3)=1/3$. The odd-squarefree class-number conjecture. If $n\equiv1\pmod{4}$, then … If…

Let $p$ and $q$ be distinct primes with $p\equiv q\equiv3\pmod{4}$, let $\alpha,\beta\geq1$, and define … Set $h^*(-p)=h(-p)$ for $p\geq7$ and $h^*(-3)=1/3$. The same-congruence cl…

Let $p$ and $q$ be distinct primes, let $\alpha,\beta\geq 1$, and define … Also set $h^*(-p)=h(-p)$ for $p\geq 7$ and $h^*(-3)=1/3$. The mixed-congruence class-number conjecture. I…

Let $H(n)$ denote the Hurwitz class number, let $\ell\geq 1$ be a fixed integer, and let $\ell_o$ denote the odd part of $\ell$. For $\nu\in\mathbb{C}$, write…

Let $n>163$ be a positive integer. For positive integers $a$ and $b$, write $n=a+b$, and let $\Omega(ab)$ denote the total number of prime factors of $ab$, counted with multiplicit…

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…

Let $m$ be a positive integer and let $d=m^2+4$ be square-free. Then $\mathbb Q(\sqrt d)$ is a real quadratic field. Yokoi's conjecture. There exist exactly six real quadratic fiel…

Let $p$ be a prime with $p\equiv1\pmod{4}$, and let $B_n$ denote the Bernoulli--Seki numbers. Define $\eth_p$ to be the least odd integer $k\geq3$ such that $p$ does not divide…