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Displaying similar documents to “Kronecker’s solution of Pell’s equation for CM fields”

On the Hilbert 2 -class field tower of some imaginary biquadratic number fields

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini, Idriss Jerrari (2021)

Czechoslovak Mathematical Journal

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Let 𝕜 = 2 , d be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and 𝕜 2 ( 2 ) its second Hilbert 2 -class field. Denote by G = Gal ( 𝕜 2 ( 2 ) / 𝕜 ) the Galois group of 𝕜 2 ( 2 ) / 𝕜 . The purpose of this note is to investigate the Hilbert 2 -class field tower of 𝕜 and then deduce the structure of G .

Bicyclic commutator quotients with one non-elementary component

Daniel Mayer (2023)

Mathematica Bohemica

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For any number field K with non-elementary 3 -class group Cl 3 ( K ) C 3 e × C 3 , e 2 , the punctured capitulation type ϰ ( K ) of K in its unramified cyclic cubic extensions L i , 1 i 4 , is an orbit under the action of S 3 × S 3 . By means of Artin’s reciprocity law, the arithmetical invariant ϰ ( K ) is translated to the punctured transfer kernel type ϰ ( G 2 ) of the automorphism group G 2 = Gal ( F 3 2 ( K ) / K ) of the second Hilbert 3 -class field of K . A classification of finite 3 -groups G with low order and bicyclic commutator quotient G / G ' C 3 e × C 3 , 2 e 6 , according to the algebraic...

A twisted class number formula and Gross's special units over an imaginary quadratic field

Saad El Boukhari (2023)

Czechoslovak Mathematical Journal

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Let F / k be a finite abelian extension of number fields with k imaginary quadratic. Let O F be the ring of integers of F and n 2 a rational integer. We construct a submodule in the higher odd-degree algebraic K -groups of O F using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of F , which is the cardinal of the finite algebraic K -group K 2 n - 2 ( O F ) .

Weight reduction for cohomological mod p modular forms over imaginary quadratic fields

Adam Mohamed (2014)

Publications mathématiques de Besançon

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Let F be an imaginary quadratic field and 𝒪 its ring of integers. Let 𝔫 𝒪 be a non-zero ideal and let p > 5 be a rational inert prime in F and coprime with 𝔫 . Let V be an irreducible finite dimensional representation of 𝔽 ¯ p [ GL 2 ( 𝔽 p 2 ) ] . We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in V already lives in the cohomology with coefficients in 𝔽 ¯ p d e t e for some e 0 ; except possibly in some few cases.

Group algebras whose groups of normalized units have exponent 4

Victor Bovdi, Mohammed Salim (2018)

Czechoslovak Mathematical Journal

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We give a full description of locally finite 2 -groups G such that the normalized group of units of the group algebra F G over a field F of characteristic 2 has exponent 4 .

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field K with 3 -class group Cl 3 ( K ) of type ( 3 , 3 ) , the structure of the 3 -class groups Cl 3 ( N i ) of the four unramified cyclic cubic extension fields N i , 1 i 4 , of K is calculated with the aid of presentations for the metabelian Galois group G 3 2 ( K ) = Gal ( F 3 2 ( K ) | K ) of the second Hilbert 3 -class field F 3 2 ( K ) of K . In the case of a quadratic base field K = ( D ) it is shown that the structure of the 3 -class groups of the four S 3 -fields N 1 , ... , N 4 frequently determines the type of principalization of the 3 -class group of K in N 1 , ... , N 4 . This...

When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin (2020)

Czechoslovak Mathematical Journal

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Let ε be an algebraic unit of the degree n 3 . Assume that the extension ( ε ) / is Galois. We would like to determine when the order [ ε ] of ( ε ) is Gal ( ( ε ) / ) -invariant, i.e. when the n complex conjugates ε 1 , , ε n of ε are in [ ε ] , which amounts to asking that [ ε 1 , , ε n ] = [ ε ] , i.e., that these two orders of ( ε ) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order [ ε 1 , ε 2 , ε 3 ] . However, there is no known similar formula for n > 3 . In the present paper, we put forward and...

The distribution of second p -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

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General concepts and strategies are developed for identifying the isomorphism type of the second p -class group G = Gal ( F p 2 ( K ) | K ) , that is the Galois group of the second Hilbert p -class field F p 2 ( K ) , of a number field K , for a prime p . The isomorphism type determines the position of G on one of the coclass graphs 𝒢 ( p , r ) , r 0 , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field K and of its p -class group Cl p ( K ) , the position of G is restricted to certain admissible branches...

On the derived length of units in group algebra

Dishari Chaudhuri, Anupam Saikia (2017)

Czechoslovak Mathematical Journal

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Let G be a finite group G , K a field of characteristic p 17 and let U be the group of units in K G . We show that if the derived length of U does not exceed 4 , then G must be abelian.

Invariance of the parity conjecture for p -Selmer groups of elliptic curves in a D 2 p n -extension

Thomas de La Rochefoucauld (2011)

Bulletin de la Société Mathématique de France

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We show a p -parity result in a D 2 p n -extension of number fields L / K ( p 5 ) for the twist 1 η τ : W ( E / K , 1 η τ ) = ( - 1 ) 1 η τ , X p ( E / L ) , where E is an elliptic curve over K , η and τ are respectively the quadratic character and an irreductible representation of degree 2 of Gal ( L / K ) = D 2 p n , and X p ( E / L ) is the p -Selmer group. The main novelty is that we use a congruence result between ε 0 -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the p -parity conjecture...