3 problems

Greenberg's generalized conjecture. The module $X(\widetilde{k})$ is pseudo-null as a $\mathbb{Z}_p\llbracket\operatorname{Gal}(\widetilde{k}/k)\rrbracket$-module.

Let $L_\infty$ be a $\mathbb{Z}_{p}^{2}$-extension of a number field $F$ containing the cyclotomic $\mathbb{Z}_{p}$-extension $F^{\mathrm{cyc}}$. Let…

Let $K$ be an imaginary quadratic field in which $p$ splits, let $\chi:\operatorname{Gal}(\overline{\mathbb{Q}}/K)\to\mathbb{Z}_p^\times$ be a finite continuous character, and let…