On Multiplicative Galois Structure Invariants

Burns, David J.

doi:10.2307/2374952

Cited by 11 publications

(4 citation statements)

References 17 publications

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“…ZqlLaQ-perfect, then the necessary computations are greatly simpli¢ed. Such extensions LaQ have been systematically studied in [14], [15] and [2] and motivated by these earlier results the following is proved in [7, ½8]. lL need no longer be ZqlLaQ-perfect, and the situation becomes considerably more complicated.…”

Section: Review Of Known Resultsmentioning

confidence: 99%

“…where the second and fourth listed isomorphisms are the obvious ones). The reduced determinant t a P E of this isomorphism is independent of the chosen splittings of each sequence (2) and (3) and of the precise choice of f and one sets w RGYE C Y tX P o Y t a P E Y P e P K 0 RGY EX It can be shown that this element is indeed independent of the precise choice of P and a. In effect, this`re¢ned Euler characteristic' construction can be used as a natural replacement for the Grothendieck^Knudsen^Mumford determinant functor (which is used systematically in [7]) in the case that G is not Abelian.…”

Section: Tamagawa Numbers and Fitting Idealsmentioning

confidence: 99%

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Bley

Burns

2001

Compositio Mathematica

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Abstract. Let LaK be a ¢nite Galois extension of number ¢elds of group G. In [4] the second named author used complexes arising from e¨tale cohomology of the constant sheaf Z to de¢ne a canonical element T OLaK of the relative algebraic K -group K 0 ZGY R. It was shown that the Stark and Strong Stark Conjectures for LaK can be reinterpreted in terms of T OLaK, and that the Equivariant Tamagawa Number Conjecture for the QG-equivariant motive h 0 Spec L is equivalent to the vanishing of T OLaK. In this paper we give a natural description of T OLaK in terms of ¢nite G-modules and also, when G is Abelian, in terms of (¢rst) Fitting ideals. By combining this description with techniques of Iwasawa theory we prove that T OLaQ vanishes for an interesting class of Abelian extensions LaQ.Mathematics Subject Classi¢cations (2000): 11R18, 11R23, 11R33.

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Section: Review Of Known Resultsmentioning

confidence: 99%

Section: Tamagawa Numbers and Fitting Idealsmentioning

confidence: 99%

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Bley

Burns

2001

Compositio Mathematica

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“…For our second result , we specialise to certain classes of fields for which this homological restriction is ameliorated and then considerably improve upon Theorem 1 . Our choice of classes of fields in this context is motivated by analogy with the results of [ 4 ] , and in turn the results of [ 4 ] were originally motivated by the recent explicit results of Fro ¨ hlich [ 17 ] on module structures of unit groups for certain special classes of real abelian fields . D E F I N I T I O N .…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Chinburg's Third Invariant for Abelian Extensions of Imaginary Quadratic Fields

Burns

Holland

1997

Proceedings of the London Mathematical Society

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In this paper the authors provide strong evidence for a conjecture of Chinburg in the context of abelian extensions N/K of quadratic imaginary fields K with group G. For abelian groups G, the conjecture specialises to the statement that Chinburg's third invariant Ω(N/K, 3)=0 ∈ Cl(Z[G]). We prove (Theorem 1) that if K has class number 1 (and some technical conditions on the conductor of N/K and its roots of unity hold) then Ω(N/K, 3) lies in the kernel group D(Z[G]). For a special class of genus extensions of prime power degree we prove (Theorem 2) the full conjecture, that Ω(N/K, 3)=0. The proof technique for Theorems 1 and 2 uses Kolyvagin's Euler systems of units applied to certain explicit groups of elliptic units, which essentially determines the difference in Galois structure between units and ideal class groups over the maximal order MG ∈ Q[G]. This is made precise by using Hecke factorisation and working in the factorisability defect class group of the category of finitely generated Z[G]‐modules, which is isomorphic to Cl(MG). For Theorem 2, the fact that the class number of N is coprime to [N:K] in such genus extensions N/K allows one to lift these results to Cl(Z[G]), after using cohomological calculations to check that (essentially) an explicit representative for a Tate sequence defining Ω (N/K, 3) has the correct extension class (which is related to the fundamental class of N/K). 1991 Mathematics Subject Classification: 11R33.

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“…The local results of Fro hlich have then been extended by Burns [4] to the case of a quadratic imaginary base field M. For more details in this direction the reader should also consult [5] and [10].…”

Section: Introductionmentioning

confidence: 99%

Computing Associated Orders and Galois Generating Elements of Unit Lattices

Bley

1997

Journal of Number Theory

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On Multiplicative Galois Structure Invariants

Cited by 11 publications

References 17 publications

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Chinburg's Third Invariant for Abelian Extensions of Imaginary Quadratic Fields

Computing Associated Orders and Galois Generating Elements of Unit Lattices

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