20 problems

Let $K/F$ be an abelian extension of number fields, let $S$ be a finite set of places containing all infinite places of $F$ and all places ramified in $K/F$, and suppose that a dis…

Bertrand–Rodriguez Villegas conjecture. There exist absolute constants $c_0>0$ and $c_1>1$ such that, for every number field $L$, every $j\in\mathbb{Z}_{>0}$, and every nonzero

A number field is called a DUG-field if every algebraic integer in it can be written as a sum of distinct units. The fields $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{5})$ have t…

Let $j_l:=(\mathbb{U}_l:\langle -1,\epsilon_l,\epsilon_l-1\rangle)$ be the unit index associated with the cubic field and units above. Louboutin's weak Ennola conjecture. For every…

Let $l\geq 3$ be an integer, and let $\epsilon_l$ generate a non-Galois totally real cubic field whose minimal polynomial is … The units $\epsilon_l$ and $\epsilon_l-1$ are excepti…

Let $K_N=\mathbb{Q}(\zeta_N)$ be the cyclotomic field of conductor $N$, where $\zeta_N$ is a primitive $N$th root of unity. A cyclotomic field is unit reducible if it has the unit-…

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$, let $X_n=2\cos(2\pi/2^{n+2})$, and let $RE_n^+=\ker({\rm N}_{n/n-1}:E_n\to E_{n-1})$. Define … The…

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$. For the norm map ${\rm N}_{n/n-1}:E_n\to E_{n-1}$, set $RE_n^+=\ker({\rm N}_{n/n-1})$, and define…

Let $\mathbb{B}_n$ be the maximal real subfield of $\mathbb{Q}(\zeta_{2^{n+2}})$, let $E_n$ be its group of units, and let ${\rm N}_{n/n-1}:E_n\to E_{n-1}$ be the norm map. Define…

Let $\mathbb B_n$ be the $n$th layer of the cyclotomic $\mathbb Z_2$-extension of $\mathbb Q$, let $E_n=\mathcal O_{\mathbb B_n}^\times$, and let … where $\mathrm{N}_{n/n-1}$ is th…

Komatsu's conjecture. For every $\epsilon\in RE_n\setminus\{\pm1\}$,

Torsion conjecture. For every $K$, $p$, and $S$,

Let $F$ be a number field, let $L_1$ be the logarithmic image of $\mathcal{O}_F^*$, and for each $k$ define $\mu_{1,k}(F)$ as the minimum of the $L^1$-norms of wedges…

Let $L$ be a number field, let $\mathcal{O}_L^*$ be its unit group, and let $\operatorname{LOG}(\mathcal{O}_L^*)$ denote the logarithmic image of the units in the real vector space…

Let $K$ be a number field with signature $(r_1,r_2)$ whose Galois closure has Galois group $S_n$, where $n$ is odd. Let $E_K$ be the unit group of $K$, and let…

A number field is called a DUG-field if every algebraic integer in it can be written as a sum of distinct units. Let $K$ be a totally complex quartic field with $\omega(K)=1$; by t…

Let $k$ be the ground field, let $T$ be the coordinate ring considered in the paper, let $T^\ast$ be its group of units, and let $n$ be the relevant degree or group order. Let $G$…

Let $k$ be the ground field, let $f$ be the polynomial defining the affine variety in the paper, and let $T$ be its coordinate ring. Write $T^\ast$ for the group of units of $T$. U…

Let $d\geq 2$ be an integer and let $m\in\mathbb{Z}\setminus\{0\}$ be such that $\sqrt[d]{m}$ has algebraic degree $d$. The order $\mathbb{Z}[\sqrt[d]{m}]$ is the subring generated…

Let $K/k$ be an abelian extension of number fields with Galois group $G$, let $S$ satisfy (St1)–(St3), and let $w$ be a place of $K$ in $S_K$ with trivial decomposition group in…