24 problems

Coates–Sujatha's conjecture B. The module $Y(E/\mathds{K}_{\infty})$ is pseudo-null as a $3bLambda_2$-module.

Let $K_infty/K$ be a $\mathbb{Z}_p$-extension, let $\mathbf{A}(rho)$ be the $p$-divisible module attached to a $p$-adic Galois representation, and let $Y(\mathbf{A}(rho)/K_infty)$…

Fine Selmer torsion conjecture. The 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 $A$ be a $g$-dimensional abelian variety defined over a number field $F$, let $F_\infty$ be a $\mathbb{Z}_p$-extension of $F$, and let $R(A/F_\infty)$ denote the fine Selmer gr…

Coates–Sujatha's conjecture. For every elliptic curve $E$ over a number field $K$, $Y(E/\textbf{K})$ is a finitely generated $\mathbb{Z}_p$-module.

Let $F$ be a number field, let $p$ be an odd prime, and let $F_\text{cyc}$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. For an elliptic curve $E$ over $F$, write…

Let $p$ be a prime number, let $F$ be a number field, let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$, and let $S$ be a finite set of primes of…

Let $K$ be the function field considered in the paper, let $L$ be a finite extension of $K$, and let $L_\mathrm{cyc}$ be its cyclotomic extension. For an elliptic curve $E/L$, let…

Let $E/\mathbb{Q}$ be an elliptic curve of Mordell–Weil rank zero, and let $a_0(E,p)$ denote the leading term appearing in Wuthrich's formula for the fine Selmer characteristic ele…

Let $\bar{A}$ be one of the coefficient objects considered in the source, with coefficient ring $\bar{R}$, and let $F_\infty/F$ be a $p$-adic Lie extension such that…

Retain the Hida-family settings of the source. Let $\mathcal{A}$ be the big Galois representation, let $F_\infty$ be a $\mathbb{Z}_{p}$-extension of a number field $F$, and write…

Let $f$ be a normalized cuspidal eigenform as in the source, let $A_f$ be the associated discrete Galois representation, and let $F_\infty$ be a $\mathbb{Z}_{p}$-extension of a num…

Let $A$ be an abelian variety defined over a number field $F$, and let $F_\infty$ be a $\mathbb{Z}_{p}$-extension of $F$ with Galois group $\Gamma=\operatorname{Gal}(F_\infty/F)$.…

Matar's conjecture. The module $Y(E/F^{\mathrm{ac}})$ is finitely generated over $\mathbb{Z}_p$.

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

Fine Selmer torsion conjecture. The module $Y(A/F_\infty)$ is torsion over $\Lambda$.

Let $\Lambda$ be the Iwasawa algebra and let $\operatorname{Sel}_0(\mathbb{Q}_\infty,E[p^\infty])^\vee$ be the Pontryagin dual of the fine Selmer group. Let…

Let $K$ be a number field, let $p$ be a prime, let $M$ be a representation of $\operatorname{G}_K$ on a finite-dimensional vector space over $\mathbb{F}_p$, and let $S$ contain the…

Let $K$ be a number field, let $p$ be a prime, and let $M$ be a representation of $\operatorname{G}_K$ on a finite-dimensional vector space over $\mathbb{F}_p$. For such a represen…

Let $K$ be a number field, let $M$ be a representation of $\operatorname{G}_K$ on a finite-dimensional vector space over $\mathbb{F}_p$, and let $\mathcal{L}$ be as in the precedin…

Let $l$ be an odd prime, let $F$ be an imaginary quadratic field, and let $\mathcal{E}$ be an elliptic curve over $F$. Let $F^{\mathrm{anti}}$ be the anticyclotomic $\mathbb{Z}_l$-…

Let $F$ be a number field with no real primes when $p=2$, let $R$ be a complete regular local ring with finite residue field of characteristic $p$, let $T$ be a finite-rank free…

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…