7 problems

Fix an odd prime $p$, a field isomorphism $\iota:\mathbb{C}\simeq\mathbb{C}_p$, a character $\psi\in\operatorname{Irr}_{\mathbb{C}_p}(\mathcal{G})$, and a finite set of places $S$…

Let $K'$ be the relevant extension, let $H'$ be its Galois group, let $I(H')$ be the augmentation ideal, and let $r_{K'}$ be the rank of the $S(K')$-unit group. Let $\theta_{H'}$ be…

Generalized Gross conjecture. The coinvariant group $X'(k^\mathrm{c}_\infty)_{\operatorname{Gal}(k^\mathrm{c}_\infty/k)}$ is finite.

Let $\psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G)$, choose $\iota:\mathbb C\simeq\mathbb C_p$, and let $r_S(\psi)$ be the corresponding complex order of vanishing. Let…

Let $\psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G)$ and let $r_S(\psi)$ be the order of vanishing at $s=0$ of the corresponding complex $S$-truncated Artin $L$-function. Gros…

With $K/F$, $G$, $V$, and $D'=\prod_{v\in V}I_{G_v,G}\subset\mathbb{Z}[G]$ as above, let $\Theta_{K,S,T}$ be the Stickelberger element. Order-of-vanishing conjecture. … This is the…

Let $(H/F,S,T,(v_i)_i,(w_i)_i)$ be as above, let $K/F$ be a finite abelian extension unramified outside $S$ with $H\subset K$ and torsion-free $\mathcal{O}_{K,S,T}^{\times}$, and p…