15 problems

Let $E/\mathbb Q$ be the number field and $\bar G$ its relevant Galois group, let $E_0=\{x\in E:\operatorname{Tr}_{E/\mathbb Q}(x)=0\}$, and let $\mu_\infty$ and $\mu_p$ be the arc…

Let $E$ be a CM elliptic curve, let $\chi$ be a character in the setting above, and assume the Dimension conjecture so that the Stark regulator ratio $A(E,\chi)$ is defined. Here…

Let $(\kappa,\lambda,\mu)\in \mathcal{W}_\textup{\bf f}^{\rm cl}$ satisfy ${\rm w}(\kappa)=2$ and ${\rm w}(\lambda)={\rm w}(\mu)=1$. Assume that $\textup{\bf f}_\kappa$ is the $p$-…

Stark's conjecture. For every automorphism $\alpha$ of $\mathbb{C}$,

Burns's conjecture. One has $\Theta(\phi)\in\mathbb{Z}[G]$, moreover $\Theta(\phi)\in\operatorname{Ann}_{\mathbb{Z}[G]}(Cl_S(K))$, and if $S'$ satisfies…

Popescu's conjecture. Assuming Hypothesis StarkHhigher, one has

Stark's conjecture over the rationals. For every $\chi\in\widehat{G}$, one has $A(\chi,\mathcal{A})\in\overline{\mathbb{Q}}$ and

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, and let $p$ be an odd prime. Let $\phi_r$, $A_{\phi_r}^S$, $\psi_r$,…

Let $F$ be the field and $n$ the integer in the source, write $n_+=\prod_{i=1}^{\nu_+}\ell_i$, let $\nu_-$ be the complementary prime-count parameter, let $h_n$ be the $n$-class nu…

Burns-style congruence conjecture.

Let $(L,S,V),(L',S',V')\in\Omega$ with $L\subset L'$, $S\subset S'$, and $V\supset V'$. Put $r=|V|$, $r'=|V'|$, and $d=r-r'$. Let $\varphi_v:L^\times\to Q_{L'/L}^1$ be the local-re…

Let $(L,S,V),(L',S',V')\in\Omega$ with $L\subset L'$, $S\subset S'$, and $V\supset V'$. Put $r=|V|$, $r'=|V'|$, and $d=r-r'$, and let $i$ be the canonical injection from the $d$-th…

Brumer–Stark conjecture. There exists $\varepsilon_{\mathfrak{a},K/k,S}\in K^{\times}$ such that

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…

Let $k$ be a totally real number field, let $k_\infty/k$ be its cyclotomic $\mathbb{Z}_p$-extension, let $L/k$ be the abelian extension and $\chi$ the character in the paper, and w…