19 problems

Stark conjecture. The Rubin–Stark element $\eta_{E/K,S(E)}^V$ should belong to the corresponding integral lattice, namely

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…

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 $\mathbb{K}$ be a number field with exactly one complex place, and consider the infinite products obtained by evaluating the geometric variants of multiple elliptic Gamma funct…

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 $p$-…

Let $F$ be the field and $G$ the group occurring above, let $\theta$, $\boldsymbol{\textstyle\bigwedge}^1_{\mathbb{C}_p[G]}$ and $\boldsymbol{e}_\chi$ be as defined in the surround…

Let $E$ be the elliptic curve associated with $f$, let $H$ be the field occurring in the triple-product setting, and let $E(H)^\rho$ denote the $\rho$-isotypic component. Let…

Let $f$ correspond to an elliptic curve $E/{\mathbb Q}$, and let $\mathcal{L}_p^g({\bf f},{\bf g},{\bf h})$ be the triple-product $p$-adic $L$-function evaluated at $(2,1,1)$. Supp…

Popescu's conjecture. Assuming Hypothesis StarkHhigher, one has

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…

Stark's conjecture. For every $r=0,-1,-2,-3,\ldots$,

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 $K\in\mathcal{K}$, let $S_K$ be the associated set of places, let $\delta_K=|\operatorname{Gal}(K/k)|$, and let $\mathcal{O}_{K,S_K}^{\times}$ be the group of $S_K$-units. Rubi…

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…