75 problems

Let $F$ be a quadratic extension of $\mathbb{Q}$ and let $K$ be a nontrivial finite abelian extension of $F$. If $F$ is real quadratic, assume that one infinite place of $F$ stays…

Finiteness conjecture. There are only finitely many primes $\ell$ for which the $\ell$-group of logarithmic classes of $K$ is nontrivial.

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…

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…

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 $d otfor all?$.

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 $p\in\mathbb N$ be prime, let $K=\mathbb Q[\sqrt{p}]$, and let $R_p$ be the order of index $p$ in $K$. An order is locally associated when it has the local association property…

Suppose $d>0$ is a square-free integer and write $d=pm$ for an odd prime $p$ and integer $m$. If $d\equiv 1\left(\bmod{\,4}\right)$, set $\delta=1$; otherwise let $\delta=2$. Let…

Stevenhagen's conjecture. As $x\to\infty$,

For every real number $B\geq 1$, let $\mathcal{R}_{B}$ be the set of ordered pairs $(\ell,K)$ such that $K$ is a quadratic field, $\ell\geq B$ is prime, and there exists an ellipti…

Kalinin's conjecture. If $p\equiv1\pmod 4$, then

The unrestricted four-squares conjecture. There are at most four distinct integer squares among the $y_k'$. If $|N_{\alpha}|$ is a prime power or a perfect square, then there are a…

Let $\overline{\mathcal{H}}_{\geq 7}(2,1)$ denote the set defined in the paper, consisting of the relevant heavenly elliptic-curve data over quadratic fields with prime parameter a…

Strong Markoff uniqueness conjecture. Each of the badly approximable irrationals $\xi$ corresponding to a Markoff triple $(x,y,z)$ has a different field of definition…

Let $d\geq 4$, set $\Delta_d=(d+1)(d-3)$, and let $K_d=\Q(\sqrt{\Delta_d})$. For a Weyl–Heisenberg line-SIC $S$ in dimension $d$, let $\fieldvec{S}$ denote its SIC field, and let…

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 $d$ range over fundamental discriminants with $|d|\leq X$. Write $\log_2 x=\log\log x$ and $\log_3 x=\log\log\log x$, and let $\theta$ and $\Theta$ be constants satisfying…

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

For prime numbers $p$ and $q$ with $p<q$ and $p\equiv q\equiv 1\pmod 4$, let $x_p,y_p,x_q,y_q$ be the unique positive integers satisfying … with $0<x_p<y_p$ and $0<x_q<y_q$. Write…

Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt{d})$, and let $\mathcal{O}_K=\mathbb{Z}[\omega]$ be the ring of integers of $K$, where … Let…

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…