scholarly journals On the capitulation of the $2$-ideal classes of the field Q(\sqrt{pq_1q_2}, i) of type (2, 2, 2)

2019 ◽  
Vol 38 (4) ◽  
pp. 127-135 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous
Keyword(s):  
Class Group ◽  
Quadratic Extensions ◽  
Genus Field

We study the capitulation of the 2-ideal classes of the field k =Q(\sqrt{p_1p_2q}, \sqrt{-1}), where p_1\equiv p_2\equiv-q\equiv1 \pmod 4  are different primes, in its three quadratic extensions contained in its absolute genus field k^{*} whenever the 2-class group of $\kk$ is of type $(2, 2, 2)$.

scholarly journals ON THE STRONGLY AMBIGUOUS CLASSES OF 𝕜/ℚ(i) WHERE $\mathds{k} = {\mathbb Q}(\sqrt{2p_{1}p_{2}, i})$

2014 ◽  
Vol 07 (01) ◽  
pp. 1450021 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous
Keyword(s):  
Infinite Family ◽  
Number Fields ◽  
Class Groups ◽  
Quadratic Extensions ◽  
Genus Field ◽  
The Absolute

We construct an infinite family of imaginary bicyclic biquadratic number fields 𝕜 with the 2-ranks of their 2-class groups are ≥ 3, whose strongly ambiguous classes of 𝕜/ℚ(i) capitulate in the absolute genus field 𝕜(*), which is strictly included in the relative genus field (𝕜/ℚ(i))* and we study the capitulation of the 2-ideal classes of 𝕜 in its quadratic extensions included in 𝕜(*).

scholarly journals Capitulation of the 2-ideal classes of type (2, 2, 2) of some quartic cyclic number fields

2019 ◽  
Vol 13 (1) ◽  
pp. 27-46
Author(s):  
Abdelmalek Azizi ◽  
Idriss Jerrari ◽  
Abdelkader Zekhnini ◽  
Mohammed Talbi
Keyword(s):  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Fundamental Unit ◽  
Class Field ◽  
Abelian Extensions ◽  
Class Field Tower

Abstract Let {p\equiv 3\pmod{4}} and {l\equiv 5\pmod{8}} be different primes such that {\frac{p}{l}=1} and {\frac{2}{p}=\frac{p}{l}_{4}} . Put {k=\mathbb{Q}(\sqrt{l})} , and denote by ϵ its fundamental unit. Set {K=k(\sqrt{-2p\epsilon\sqrt{l}})} , and let {K_{2}^{(1)}} be its Hilbert 2-class field, and let {K_{2}^{(2)}} be its second Hilbert 2-class field. The field K is a cyclic quartic number field, and its 2-class group is of type {(2,2,2)} . Our goal is to prove that the length of the 2-class field tower of K is 2, to determine the structure of the 2-group {G=\operatorname{Gal}(K_{2}^{(2)}/K)} , and thus to study the capitulation of the 2-ideal classes of K in all its unramified abelian extensions within {K_{2}^{(1)}} . Additionally, these extensions are constructed, and their abelian-type invariants are given.

Hybrid subconvexity for class group 𝐿-functions and uniform sup norm bounds of Eisenstein series

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Eisenstein Series ◽  
Class Group ◽  
Heegner Points ◽  
Quadratic Extensions ◽  
Norm Bound ◽  
Subconvexity Bounds

AbstractIn this paper, we study hybrid subconvexity bounds for class group 𝐿-functions associated to quadratic extensions K/\mathbb{Q} (real or imaginary). Our proof relies on relating the class group 𝐿-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E(z,1/2+it)\ll_{\varepsilon}y^{1/2}(\lvert t\rvert+1)^{1/3+\varepsilon}, y\gg 1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.

scholarly journals On the rank of the first radical layer of a p-class group of an algebraic number field

1999 ◽  
Vol 156 ◽  
pp. 85-108
Author(s):  
Hiroshi Yamashita
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sylow Subgroup ◽  
Class Group ◽  
Cyclotomic Field ◽  
Artinian Ring ◽  
Algebraic Number Field ◽  
Central Extensions ◽  
Genus Field ◽  
Global And Local

Let p be a prime number. Let M be a finite Galois extension of a finite algebraic number field k. Suppose that M contains a primitive pth root of unity and that the p-Sylow subgroup of the Galois group G = Gal(M/k) is normal. Let K be the intermediate field corresponding to the p-Sylow subgroup. Let = Gal(K/k). The p-class group C of M is a module over the group ring ZpG, where Zp is the ring of p-adic integers. Let J be the Jacobson radical of ZpG. C/JC is a module over a semisimple artinian ring Fp. We study multiplicity of an irreducible representation Φ apperaring in C/JC and prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic field M such that the minus part of C is cyclic as a ZpG-module and a CM-field M such that the plus part of C vanishes for odd p.To show the formula, we apply theory of central extensions of algebraic number field and study global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect to M/K.

Principalization of 2-class groups of type (2, 2, 2) of biquadratic fields ${\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$

2015 ◽  
Vol 11 (04) ◽  
pp. 1177-1215 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous ◽  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Nilpotency Class ◽  
Class Groups ◽  
Class Field ◽  
Abelian Extensions ◽  
Biquadratic Fields

Let p1 ≡ p2≡ -q ≡ 1 (mod 4) be different primes such that [Formula: see text]. Put d = p1p2q and [Formula: see text], then the bicyclic biquadratic field [Formula: see text] has an elementary abelian 2-class group, Cl2(𝕜), of rank 3. In this paper, we study the principalization of the 2-classes of 𝕜 in its 14 unramified abelian extensions 𝕂j and 𝕃j within [Formula: see text], that is the Hilbert 2-class field of 𝕜. We determine the nilpotency class, the coclass, generators and the structure of the metabelian Galois group [Formula: see text] of the second Hilbert 2-class field [Formula: see text] of 𝕂. Additionally, the abelian type invariants of the groups Cl2(𝕂j) and Cl2(𝕃j) and the length of the 2-class tower of 𝕜 are given.

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