188 problems

Let $p$ be an odd prime and let $t$ be a positive integer. A number field $K$ is $p$-rational if, for the set $S$ of primes of $K$ lying above $p$, the Galois group of the maximal…

Gras's conjecture. For $p\gg 0$, one has

Brumer's conjecture. If $G$ is abelian, then

Let $\theta_n$, for $n\geq 2$, be the unique root in $(0,1)$ of $-1+x+x^n$, and call a real algebraic integer greater than $1$ a Salem number when it has the usual Salem-number pro…

Coates–Sinnott conjecture. If $G$ is abelian, then

Schinzel–Zassenhaus conjecture. There exists a constant $c>0$ such that, if $\alpha$ is not a root of unity, then

Let $m$ and $d$ be positive integers, and let $K$ range over all number fields of degree $d$. The $m$-rank of the class group of $K$ is the largest integer $r$ such that…

Let $L/K$ be a finite weakly ramified Galois extension of number fields, with $G=\operatorname{Gal}(L/K)$, ring of integers $\mathcal{O}_L$, and different $\mathcal{D}_{L/K}$. When…

Let $F$ be a real quadratic field, let Z_\\A(s) be the difference of two ray class partial zeta functions associated with a ray class, and let $\exp(-Z_\A'(0))$ be the correspond…

Let $F_n$ be the cyclotomic field in question, let ${\mathbb A}(F_n)$ be its ring of adeles, and let $K_{p^{n+1}-1}$ be the subgroup … of $GL(1,{\mathbb A}(F_n))$, where the produc…

Lower-bound conjecture. We conjecture that the following bounds hold: for $q\in(1,2)$,

Chinburg's $\Omega(3)$ conjecture modulo the kernel group.

Nonvanishing determinant conjecture. $\det(N_\ell)\neq 0$ for every odd prime $\ell$.

Let $\beta>1$ be the root of … where $m,n\in\mathbb N$ and $m\geq n+2\geq 3$. Let $\operatorname{Fin}(-\beta)$ denote the set of numbers having finite $(-\beta)$-expansions. Ito–Sa…

Probabilistic subgroup-generation conjecture. Even though one does not generally know that $H=K_S$, the kernel and cokernel of $\phi$ should be the unit group and ideal class group…

Fix an integer $d \geq 3$. For a totally real number field $K$ of degree $d$, let $\operatorname{disc}(K)$ denote its discriminant, and let an ideal class mean a class of fractiona…

Let $\beta$ be a Pisot number with Property (F), and let $\alpha$ be a conjugate of $\beta$. For $x\in\mathbb{R}$, let $\phantom{}_{\alpha}\#(x)$ denote the number of different…

Let $N$ be a prime level, let $\chi$ be the character used to define $h_\chi$, and let $B_{m,\chi}$ denote the $m$-th generalised Bernoulli number attached to $\chi$. Write … For e…

Algebraicity–multiplier group conjecture. A quasiperiodic flow $\psi$ is $F$-algebraic if and only if $M_\psi$ is a finite index subgroup of ${\mathfrak o}_F^*$.

Let $n\geqslant 3$ and let $\Lambda\subset\mathbb{R}^n$ be an $n$-dimensional lattice with positive norm minimum $N(\Lambda)>0$. A lattice is called algebraic when it is similar, m…

Power integral basis removal conjecture. The requirement that $K$ have a power integral basis can be cancelled: the displayed set is empty or has at least $2^m$ elements.

Let $T^n$ be the $n$-torus, let $\frac{d}{dt}x=\boldsymbol{\theta}$ denote a quasiperiodic flow on $T^n$, and let $M_\psi$ be its multiplier group. A flow is algebraic when its fre…

Let $p$ be an odd prime splitting completely in $k$, let $K/k$ and the other notation be as in the paper, and let $G=\operatorname{Gal}(K/k)$. Let $\eta^+_{K/k}$ be the image of th…

Let $K^+/k$ be the totally real subextension under consideration, let $G^+=\operatorname{Gal}(K^+/k)$, let $S_0=S_{\rm ram}(K/k)$, and let $U_{S_0}(K^+)$ be the group of global…

Let $k$ be a totally real number field, let $p$ be an odd prime splitting completely in $k$, and let $K$ be as in the paper. Write $G=\operatorname{Gal}(K/k)$ and let $\delta(K/k)$…