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...

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...

Some results on the cofiniteness of local cohomology modules

Sohrab Sohrabi Laleh, Mir Yousef Sadeghi, Mahdi Hanifi Mostaghim (2012)

Czechoslovak Mathematical Journal

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Let R be a commutative Noetherian ring, 𝔞 an ideal of R , M an R -module and t a non-negative integer. In this paper we show that the class of minimax modules includes the class of 𝒜ℱ modules. The main result is that if the R -module Ext R t ( R / 𝔞 , M ) is finite (finitely generated), H 𝔞 i ( M ) is 𝔞 -cofinite for all i < t and H 𝔞 t ( M ) is minimax then H 𝔞 t ( M ) is 𝔞 -cofinite. As a consequence we show that if M and N are finite R -modules and H 𝔞 i ( N ) is minimax for all i < t then the set of associated prime ideals of the generalized local cohomology...