42 problems

Let $(0,0)$ be an isolated singular point of an algebraic curve over a number field $k$, with local ring $\mathcal{O}=k[[x,y]]/(f)$ and multiplicity $m$. Let $C$ be the nonsingular…

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 $k$ 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 $\delta(K/k)$…

Let $t=(d,r,Q)$ be an admissible tuple with associated real quadratic field $K$, let $d=d_{j,m}$, and let $f$ be the conductor of $Q$. Let $s=(d,r,Q,G,g)$ be a fiducial datum. Deno…

Let $H/F$ be an abelian extension of number fields, let $W$ be the number of roots of unity in $H$, and let $S$ be a finite set of places of $F$ containing the ramified and infinit…

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 $K$ be an imaginary quadratic field with discriminant $d_K$, suppose $(5)=\wp_5^2$, let $\Sigma=\textsf{K}_1$, and let $\rho\in\Sigma$. Let…

Let $d$ be an integer with quadratic discriminant $d_k=\prod_{p\mid d}p^*$, where $p^*=(-1)^{(p-1)/2}p$ for odd $p$, and let $2^*\in\{-4,8,-8\}$. Let $\textsf{K}_{\wp_5'}$ be the i…

Let $K$) be an imaginary quadratic field with discriminant $d_K$, let $(2)=\mathfrak p_1\mathfrak p_2$, let $h(d_K)$ denote the class number, and let $\textsf{K}_1$ denote the rele…

Fix open subgroups $U^p\subseteq G(\mathbb{A}^{p,\infty})$ and $U_p\subseteq G(\mathbb{Q}_p)$ containing $G(\mathbb{Q}_p)^+$, and put $U=U^pU_pG(\mathbb{R})^+$. For a coset…

Let $L/H/F$ be a tower in which $F$ is totally real and $H$ and $L$ are finite abelian CM extensions of $F$, with $L$ containing $H$. Write $\mathcal{G}=\operatorname{Gal}(L/F)$,…

Let $L/K$ be any cyclic $p$-extension of degree $p^N$, with $N \geq 1$, and Galois group $G$. Assume that $L/K$ is ramified at $r \geq 1$ places of $K$, totally ramified in $L/K$.…

Let $K$ be any real number field and let ${\mathcal H}_K$ be its $p$-class group, of exponent $p^{e(K)}$. For any prime number $\ell \equiv 1 \pmod {2p^N}$, with $N \geq e(K)$, let…

Strong cyclotomic conjecture. The characteristic polynomial of the $\Lambda$-module $\mathcal C^{T_K}_{S_K}(K_\infty)$ is not divisible by $\omega=\gamma-1$. Equivalently, its fixe…

Brumer–Stark conjecture. There exists $u_T\in\mathcal{U}_{\mathfrak{p}}$ such that $u_T\equiv1\pmod T$ and, for every $\sigma\in G$,

Let $X$ be a smooth quasi-projective variety over the local field $k$. Write … for the tame reciprocity map, where $E^{\rm t}(X)$ and $J^{\rm t}(X)$ are the tame class group and ta…

Let $H/F$ be a finite abelian CM-extension, let $p$ be an odd prime, and put $r=\#S'_f-1$. Let $\operatorname{Ker}(\theta_V)_H$ be the theta-kernel module, let…

Non-commutative Moore sequence conjecture. One expects an isomorphism

Let $F$, $H$, $S$, $T$, and $\mathfrak{p}$ be as above, and let $L/F$ be a finite abelian CM extension containing $H$ and unramified outside…

Let $L$ be a non-Galois totally real cubic field with ramified Galois closure $N$, conductor $f>1$ over a real quadratic field $K$, and type $\alpha$, with $U_N=U_0$. Let…

Extension-frequency conjecture. Precisely one half of unramified $[2,2]$-extensions of cyclic cubic fields extend to unramified $\langle8,4\rangle$-extensions.

Wide realization conjecture. Every finite $2$-select $2$-group arises as the wide tower group of some cyclic cubic field.

Narrow realization conjecture. Every finite $2$-special $2$-group arises as the narrow tower group of some cyclic cubic field.

Mayer's conjecture. Non-Galois totally real cubic fields with these properties should exist in each of the following situations: type $\alpha_1$ with…

Scholz's conjecture. There should exist such fields $L$ for which either $f=1$, $\varrho_3(K)=2$, the complete $3$-elementary class group of $K$ capitulates in $N$, and $U_N=U_0$,…