Displaying similar documents to “On the classgroups of imaginary abelian fields”

Iwasawa theory for elliptic curves over imaginary quadratic fields

Massimo Bertolini (2001)

Journal de théorie des nombres de Bordeaux

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Let E be an elliptic curve over ℚ , let K be an imaginary quadratic field, and let K ∞ be a ℤ p -extension of K . Given a set Σ of primes of K , containing the primes above p , and the primes of bad reduction for E , write K Σ for the maximal algebraic extension of K which is unramified outside Σ . This paper is devoted to the study of the structure of the cohomology groups H i ( K Σ / K ∞ , E p ∞ ) for i = 1 , 2 , and of the p -primary Selmer group Sel p ∞ ( E / K ∞ ) , viewed as discrete modules over the Iwasawa algebra of K ∞ / K . ...

An annihilator for the p -Selmer group by means of Heegner points

Massimo Bertolini (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let E / Q be a modular elliptic curve, and let K be an imaginary quadratic field. We show that the p -Selmer group of E over certain finite anticyclotomic extensions of K , modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic Z p -extension of K . This refines in the current contest a result of [1].

Class groups of abelian fields, and the main conjecture

Cornelius Greither (1992)

Annales de l'institut Fourier

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This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p = 2 , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ -parts of p -class groups of abelian number fields: first for relative class groups of real fields (again including the case p = 2 ). As a consequence, a generalization of the Gras conjecture...

Circular units of real abelian fields with four ramified primes

Vladimír Sedláček (2017)

Archivum Mathematicum

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In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct ℤ -bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results...

Capitulation and transfer kernels

K. W. Gruenberg, A. Weiss (2000)

Journal de théorie des nombres de Bordeaux

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If K / k is a finite Galois extension of number fields with Galois group G , then the kernel of the capitulation map C l k → C l K of ideal class groups is isomorphic to the kernel X ( H ) of the transfer map H / H ' → A , where H = Gal ( K ˜ / k ) , A = Gal ( K ˜ / K ) and K ˜ is the Hilbert class field of K . H. Suzuki proved that when G is abelian, | G | divides | X ( H ) | . We call a finite abelian group X a transfer kernel for G if X ≅ X ( H ) for some group extension A ↪ H ↠ G . After characterizing transfer kernels in terms of integral representations of G , we show that X is a transfer...

Relative Galois module structure of integers of abelian fields

Nigel P. Byott, Günter Lettl (1996)

Journal de théorie des nombres de Bordeaux

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Let L / K be an extension of algebraic number fields, where L is abelian over ℚ . In this paper we give an explicit description of the associated order 𝒜 L / K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L , is then isomorphic to 𝒜 L / K . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that 𝒜 L / K is the maximal order if L / K is a cyclic and totally wildly ramified extension which is linearly disjoint to ℚ ( m ' ) / K , where m ' is the conductor...

Artinian cofinite modules over complete Noetherian local rings

Behrouz Sadeghi, Kamal Bahmanpour, Jafar A'zami (2013)

Czechoslovak Mathematical Journal

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Let ( R , 𝔪 ) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R -module. In this paper it is shown that if 𝔭 is a prime ideal of R such that dim R / 𝔭 = 1 and ( 0 : M 𝔭 ) is not finitely generated and for each i ≥ 2 the R -module Ext R i ( M , R / 𝔭 ) is of finite length, then the R -module Ext R 1 ( M , R / 𝔭 ) is not of finite length. Using this result, it is shown that for all finitely generated R -modules N with Supp ( N ) ⊆ V ( I ) and for all integers i ≥ 0 , the R -modules Ext R i ( N , M ) are of finite length, if and only if, for all finitely generated R -modules...