123 problems

Let $p$ be a prime satisfying … For an elliptic curve $E/\mathbb Q$, write $E(\mathbb Q(\sqrt{-p}))_{\mathrm{tors}}$ for its torsion subgroup over $\mathbb Q(\sqrt{-p})$. Existence…

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…

Chapman's conjecture.

Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt d)$, and let $\mathcal{O}_K$ be its ring of integers. Define … and let $\varepsilon>1$ be the fundamental unit…

Let $k \geq 1$ be an integer and let $d$ be an integer with $d \not\equiv 2 \bmod 4$. For a prime $p$, consider the quadratic field $\mathbb{Q}(\sqrt{dp})$. Cohn–Lagarias conjectur…

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

Let $K/\mathbb{Q}$ be a quadratic field, let $f\in K[z]$ be a quadratic polynomial, and let $\operatorname{PrePer}(f,K)$ denote its set of $K$-rational preperiodic points. Doyle's…

Let $K$ be a quadratic number field, and write $\operatorname{Cl}(K)[2^{\infty}]$ for the $2$-primary part of its ideal class group. Gerth's conjecture. The distribution of…

Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt d)$, and let $\mathcal{O}_K$ be its ring of integers. Write the fundamental unit $\varepsilon>1$ as…

Let $p$ be an odd prime, and let the $p$-class tower group of a field $K$ be the Galois group of its maximal unramified $p$-extension, equivalently the pro-$p$ completion of…

Let $a$, $b$ and $d$ be positive integers such that $d$ is not a square. Set … and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ with $t,u$ po…

Let $D$ range over fundamental positive discriminants without prime factors congruent to $3$ modulo $4$. Define … Stevenhagen's conjecture. The density of those $D$ for which the n…

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 $K=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field, and let $C_1$ be the cone whose elements have angle between $\pi/36$ and $2\pi/36$. Let $P_M(-d)$ count primes less t…

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 $p>13$.…

McMullen's Arithmetic Chaos conjecture. There is a compact subset $\mathcal{Y}\subset\mathcal{X}$ such that, for every real quadratic field $K$, the set of closed geodesics defined…

Let $g(u)$ be a suitable weight function, and let $\mathcal F$ be the family of real Dirichlet $L$-functions $L(s,\chi_d)$ associated with negative fundamental discriminants $d<0$.…

Let $K=\mathbb{Q}(\sqrt{\mathfrak{d}})$ range over real quadratic fields, where $\mathfrak{d}\geq 2$ is square-free, and let $\mathcal{O}_K$ denote the ring of integers of $K$. The…

Let $d$ range over discriminants satisfying the Pellian condition, namely those for which every odd prime divisor of $d$ is congruent to $1$ modulo $4$. Consider the discriminants…

Let $p$ be a prime, let $k'$ be the quadratic field and $\varepsilon'$ a unit of $k'$ occurring in the algorithm, and let $h_{k'}$ denote its class number. Let $\eta_1$ and…

The first tower-length conjecture. If

Conjecture. $M'_{\mathrm{even}}(d)\leq 2$ if and only if $d\in\{2,6,10,14,22,26,30,38,42,62,110,122,182,278,362,398\}$.

Let $d otfor all?$.

Let $d otfor all?$.

For squarefree $D>1$, let $\mathcal{O}$ be the ring of integers in $\mathbb{Q}(\sqrt{D})$, and let $\mathcal{S}_D$ consist of the elements $u=A+B\sqrt{D}$ with $A,B\in\mathbb{Z}$ s…