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

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.

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...

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...

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...

On the compositum of all degree d extensions of a number field

Itamar Gal, Robert Grizzard (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the compositum k [ d ] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [ d ] contains all subextensions of degree less than d if and only if d 4 . We prove that for d > 2 there is no bound c = c ( d ) on the degree of elements required to generate finite subextensions of k [ d ] / k . Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d , but that one can take c = d when d is prime. This question was inspired by work of...

On the structure of the Galois group of the Abelian closure of a number field

Georges Gras (2014)

Journal de Théorie des Nombres de Bordeaux

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From a paper by A. Angelakis and P. Stevenhagen on the determination of a family of imaginary quadratic fields K having isomorphic absolute Abelian Galois groups A K , we study any such issue for arbitrary number fields K . We show that this kind of property is probably not easily generalizable, apart from imaginary quadratic fields, because of some p -adic obstructions coming from the global units of K . By restriction to the p -Sylow subgroups of A K and assuming the Leopoldt conjecture we...