Related papers: On some $p$-adic Galois representations and form c…

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_\mathfrak{n}$ be the ray class field of $K$ modulo $\mathfrak{n}=N\mathcal{O}_K$. By using the…

Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to…

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary…

Let $\ell$ be an odd prime and $d$ a positive integer. We determine when there exists a degree-$d$ number field $K$ and an elliptic curve $E/K$ with $j(E)\in\mathbb{Q}\setminus\{0,1728\}$ for which $E(K)_{\mathrm{tors}}$ contains a point of…

For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over…

The main purpose of this paper is to extend results on isomorphism types of the abelianized absolute Galois group $\mathcal G_K^{ab}$, where $K$ denotes imaginary quadratic field. In particular, we will show that if the class number $h_K$…

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 $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $h_E$ be the Weber function on certain elliptic curve $E$ with complex multiplication by $\mathcal{O}_K$. We show that if $N$ ($>1$) is an integer…

Let $K/k$ be a pro-$p$-extension over a number field $k$ whose Galois group is finitely generated and $k_0\subseteq k_1\subseteq\cdots\subseteq k_n\subseteq\cdots$ an ascending sequence of intermediate fields of $K/k$ such that $k_n/k$ is…

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a nontrivial integral ideal $\mathfrak{m}$ of $K$, let $K_\mathfrak{m}$ be the ray class field modulo $\mathfrak{m}$. By using…

We study the action of the Galois group $G$ of a finite extension $K/k$ of number fields on the points on an elliptic curve $E$. For an odd prime $p$, we aim to determine the structure of the $p$-adic completion of the Mordell-Weil group…

Let $K/\mathbb{Q}$ be a finitely generated field of characteristic zero and $X/K$ a smooth projective variety. Fix $q\in\mathbb{N}$. For every prime number $\ell$ let $\rho_\ell$ be the representation of $\mathrm{Gal}(K)$ on the \'etale…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a $\mathbb{Z}_p$-extension of $K$. We prove that $E(L)_{\text{tors}}=E(K)_{\text{tors}}$ when $p>5$. It enables…

Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we…

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_E for a…

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers…

Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal…

Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's…

Let $\ell$ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb{Q}_{\ell}$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb{Q}_{\ell}(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree…

For a prime \(p\ge 2\) and a number field K with p-class group of type (p,p) it is shown that the class, coclass, and further invariants of the metabelian Galois group \(G=Gal(F_p^2(K) | K)\) of the second Hilbert p-class field \(F_p^2(K)\)…