Related papers: Notes on the fine Selmer groups

We study the groups in the unit filtration of a finite abelian extension K of the field of p-adic numbers. We determine explicit generators of these groups as modules over the pro-p group ring of the Galois group of K over the p-adic…

For fields F of characteristic not p containing a primitive $p$th root of unity, we determine the Galois module structure of the group of $p$th-power classes of K for all cyclic extensions K/F of degree p.

Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2 \in S_k(\Gamma_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $\omega_2: G_{\mathbb Q} \rightarrow {\mathbb…

Let E be a nonarchimedean local field with residue characteristic l, and suppose we have an n-dimensional representation of the absolute Galois group G_E of E over a reduced complete Noetherian local ring A with finite residue field k of…

Greenberg examined the local behavior of Iwasawa invariants as functions on the the set of all $\mathbb{Z}_p$-extensions of a number field $F$. Kleine later extended these ideas to explore the variation of Iwasawa invariants in the context…

Let T be a free Z_p-module of finite rank equipped with a continuous Z_p-linear action of the absolute Galois group of a number field K satisfying certain conditions. In this article, by using a Selmer group corresponding to T, we give a…

The P\'{o}lya group of an algebraic number field is a particular subgroup of the ideal class group. This article provides an overview of recent results on P\'{o}lya groups of number fields, their connection with the ring of integer-valued…

We investigate the Galois group $G_S(p)$ of the maximal $p$-extension unramified outside a finite $S$ of primes of a number field in the (tame) case, when no prime dividing $p$ is in $S$. We show that the cohomology of $G_S(p)$ is 'often'…

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur…

Let $p\geq 5$ be a prime number. We consider the Iwasawa $\lambda$-invariants associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. Let $g$ be a $p$-ordinary cuspidal…

Let $G$ be a finite $p$-group. We construct a $G$-extension $K/k$ of number fields such that the $p$-adic completion of the unit group of $K$ has a prescribed $\mathbb{Z}_p[G]$-module structure, up to free direct summands.

This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the…

We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely…

Let $p$ be a prime number, let $K$ be a $p$-field (a local field with finite residue field of characteristic $p$), let $L$ be a finite galoisian tamely ramified extension of $K$, and let $G=\mathrm{Gal}(L|K)$. Suppose that $L$ is split over…

In this article, we study the pseudo-isomorphism class of the dual fine Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation ring $\Lambda$ is isomorphic to the ring of formal power series. By using the "Kolyvagin…

Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each…

The p-class tower $F_p^\infty(k)$ of a number field k is its maximal unramified pro-p extension. It is considered to be known when the p-tower group, that is the Galois group $G:=Gal(F_p^\infty(k)/k)$, can be identified by an explicit…

For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the…

Let $p$ be an odd prime number and $f$ a modular form. We consider the $\mathbb{F}_p$-valued Galois representation $\bar{\rho}_f$ attached to $f$ and its twist $\bar{\rho}_{f, D}$ by the quadratic character $\chi_D$ corresponding to a…

Let F denote an unramified extension of the cyclotomic extension of Q_p by (p^n)th roots of unity, for an odd prime p. We determine the conductors of those Kummer extensions of F of degree dividing p^n which are Galois over the maximal…