Related papers: Singular integers and p-class group of cyclotomic …

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'…

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…

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

Recently the second author has associated a finite $\F_q[T]$-module $H$ to the Carlitz module over a finite extension of $\F_q(T)$. This module is an analogue of the ideal class group of a number field. In this paper we study the Galois…

Each p-ring class field K(f) modulo a p-admissible conductor f over a quadratic base field K with p-ring class rank r(f) mod f is classified according to Galois cohomology and differential principal factorization type of all members of its…

Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$…

In this paper we consider the problem of classifying the isomorphism classes of extensions of degree pk of a p-adic field, restricting to the case of extensions without intermediate fields. We establish a correspondence between the…

We prove that the Krull-Schmidt decomposition of the Galois module of the $p$-adic completion of algebraic units is controlled by the primes that are ramified in the Galois extension and the $S$-ideal class group. We also compute explicit…

Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit…

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ be a primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is a pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$.…

Given an odd prime number $p$ and an imaginary quadratic field $K$, we establish a relationship between the $p$-rank of the class group of $K$, and the classical $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $K$.…

Let $E/\mathbb{Q}$ be an elliptic curve and let $p\ge 5$ be a prime of good supersingular reduction. We generalize results due to Meng Fai Lim proving Kida's formula and integrality results for characteristic elements of signed Selmer…

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.

Let $p$ be an odd prime and $L/K$ a $p$-adic Lie extension whose Galois group is of the form $\mathbb{Z}_p^{d-1}\rtimes \mathbb{Z}_p$. Under certain assumptions on the ramification of $p$ and the structure of an Iwasawa module associated to…

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O_k of k, which, together with the degree [K:k] of the extension determines the O_k-module structure of O_K. We call rt(k,G) the classes which are…

We determine the exact values of the linear complexity of 2p-periodic quaternary sequences over Z_4 (the residue class ring modulo 4) defined from the generalized cyclotomic classes modulo 2p in terms of the theory of of Galois rings of…

Let $p\ge 5$ be a prime number and $E/\mathbf{Q}$ an elliptic curve with good supersingular reduction at $p$. Under the generalized Heegner hypothesis, we investigate the $p$-primary subgroups of the Tate--Shafarevich groups of $E$ over…

Let $p$ be a prime and let $F$ be a number field. Consider a Galois extension $K/F$ with Galois group $H\rtimes \Delta$ where $H\cong \mathbb{Z}_p$ or $\mathbb{Z}/p^d\mathbb{Z}$, and $\Delta$ is an arbitrary Galois group. The subfields…

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ an odd prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$. In a previous article the author conjectured that the fine $p^{\infty}$-Selmer…

For a prime p, we study the Galois groups of maximal pro-$p$ extensions of imaginary quadratic fields unramified outside a finite set $S$, where $S$ consists of one or two finite places not lying above $p$. When $p$ is odd, we give explicit…