5 problems

Let $p$ be a prime number and $k$ a number field. Let $\widetilde{k}$ be the maximal multiple $\mathbb{Z}_p$-extension field of $k$, let $L_{\widetilde{k}}$ be its maximal unramifi…

For general $N>1$, let $X^+$ denote the plus part of the relevant cyclotomic Iwasawa module. Greenberg's conjecture. The module $X^+$ is finite. This is a finiteness form of Greenb…

Let $bK/Q$ be an arbitrary Galois extension containing the $p$-th roots of unity, and let $H/K_{\infty}$ be the maximal unramified $p$-abelian extension of the cyclotomic $Z_p$-exte…

Let $K$ be a number field, and let $L'_{\infty}$ and $T_{\infty}$ be the fields appearing in the source's Iwasawa-theoretic construction. Generalized Greenberg conjecture. For an a…

Combined Iwasawa–Greenberg conjecture. The following groups have orders bounded independently of $n$ and $i$: (i) the $\ell$-kernels $W\!K_{4i}(F_n)$; and (ii) the imaginary compon…