9 problems

Let $K$ be a number field, let $K_{\infty,\ell}=K\cdot\mathbb{Q}_{\infty,\ell}$ be its cyclotomic $\mathbb{Z}_\ell$-extension, and let $X$ be a curve of genus at least $2$ over…

Let $K$ be a number field, let $K_{\infty,\ell}=K\cdot\mathbb{Q}_{\infty,\ell}$ be its cyclotomic $\mathbb{Z}_\ell$-extension, and let $A$ be an abelian variety over…

For a number field $E$, let $E^{\rm cyc}$ be its cyclotomic $\mathbb{Z}_p$-extension, let $\Gamma_E={\rm Gal}(E^{\rm cyc}/E)$, and let $A_S(E^{\rm cyc})$ be the relevant Iwasawa mo…

Coates–Sujatha conjecture. The module $Y(A/F^{\mathrm{cyc}})$ is finitely generated over $\mathbb{Z}_p$.

Cyclotomic fine Tate–Shafarevich growth conjecture. One has

Cyclotomic Mordell–Weil conjecture. The group $A(F^{\mathrm{cyc}})$ is a finitely generated abelian group.

Greenberg's conjecture. The characteristic polynomial of the $\Lambda$-module $\mathcal C^T_S(K_{\infty})$ is not divisible by $\omega$. Equivalently, its fixed submodule is finite…

Let $F$ be a number field, let $R$ be the coefficient ring, let $T$ be the Galois module considered in the paper, let $F^{\mathrm{cyc}}$ be the cyclotomic $\mathds{Z}_{p}$-extensio…

Let $K$ be a number field, let $\ell$ be a prime, and put $E=K(\ell)=K(\mu_{\ell^{\infty}})$. Let $C$ be an absolutely irreducible smooth projective curve over $E$, and write…