37 problems

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…

Rubin–Stark conjecture. The Rubin–Stark element satisfies

Brauer–Siegel conjecture. Then

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…

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

The Möbius-inversion class-number conjecture. One has

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 $K$ be a number field and let $K^{\text{cyc}}$ be the cyclotomic $\mathbf{Z}_p$-extension of $K$. The Iwasawa $\mu$-invariant is the coefficient governing the exponential term…

Let $q$ be a prime, let $h_1(q)$ be the relative class number of the cyclotomic field $\mathbb Q(\zeta_q)$ over its maximal real subfield, and define … The Kummer conjecture. As…

Let $q$ be a prime number, and let $h_1(q)$ denote the relative class number. Define … The ratio $R(q)$ is called the Kummer ratio. Kummer's conjecture. As $q$ tends to infinity, ……

For an imaginary quadratic field $K$, let $D_K$ and $h(K)$ denote its discriminant and class number, respectively. Gauss's class-number conjecture. Then … This conjecture asserts t…

Let $m$ be a positive integer, let $\mathbb{Q}(\zeta_m+\zeta_m^{-1})$ be the maximal real subfield of the $m$th cyclotomic field, and let $h^+(m)$ denote its class number. Polynomi…

Let $n$ range over positive integers such that $p=4n-1$ is prime, and let $M$, $R_m$, and $Q_m$ be the quantities defined earlier in the paper. Asymptotic floor-sum conjecture. … M…

Let $n$ range over positive integers such that $p=4n-1$ is prime, and let $J_n$ be the quantity defined earlier in the paper. Asymptotic conjecture for $J_n$. … The paper presents…

Let $p$ be a prime such that $p\equiv 3\pmod 4$. Let $h(-8p)$ and $h(8p)$ denote the class numbers associated with the discriminants $-8p$ and $8p$, respectively, and let $\Psi$ be…

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…

Let $l>7$ be prime with $l\equiv1,6\pmod7$. Let $\hat H_{7,l}(x)$ be the Hasse-invariant polynomial, and let $N_2$ count irreducible quadratic factors $x^2+ax+b$ dividing it modulo…

Let $l\neq 2,3,7$ be prime. Let $\hat H_{7,l}(x)$ be the Hasse-invariant polynomial of the Tate normal form $E_7$ over $\mathbb F_l$, and let $h(-l)$ be the relevant class number.…

Generalized Iizuka conjecture. For any odd integer $n\geq 3$ and any integer $m\geq 1$, there is an infinite family of $m+1$ successive imaginary or real quadratic fields of this f…

Let $p_i$) be the $i$-th prime, let $\mathfrak{P}=\{\mathbb{Q}(\sqrt{p})\mid p\text{ prime}\}$, and define … Assume these probabilities exist for all positive odd integers $q$. Wri…

Vandiver's conjecture. $X_\infty^{+}=0$. Equivalentemente, $p$ no divide el número de clases del subcampo $\mathbb{Q}(\mu_p)^{+}$ de $\mathbb{Q}(\mu_p)$ fijado por la conjugación c…

Hooley's conjecture. For every $\alpha>0$, there exists a constant $b(\alpha)>0$ such that