Tate sequences and Fitting ideals of Iwasawa modules

Greither, Cornélius; Kurihara, Masato

doi:10.1090/spmj/1428

Cited by 10 publications

(37 citation statements)

References 10 publications

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“…A situation similar to this arose in work [17], completed in [19], of Kurihara, Tokio (the name has nothing to do with the city Tokyo) and the author. We considered an abelian extension K/k of totally real fields with group G and the Iwasawa module X S discussed before, which is the Galois group of the maximal abelian S-ramified p-extension of K ∞ .…”

Section: About 4-sequences and Cohomologymentioning

confidence: 75%

“…Second Scenario [12,17] We adapt and simplify some notation from these papers for the present purpose.…”

Section: Example Scenariosmentioning

confidence: 99%

“…The second issue is treated in [17] completed by [19]; the issue of eliminating the major cheat is dealt with in [20]. We will discuss all this briefly in the final section.…”

Section: Example Scenariosmentioning

confidence: 99%

“…In terms of class field theory, this is a projective limit not of class groups but of ray class groups; in the totally real case it is the right object to study, except for the fact that the set of places needs to contain not only the padic places (forced by Iwasawa theory) but all places ramified in K/k. In [17] the technique of exact 4-sequences is used to determine the Fitting ideal of X S over the appropriate Iwasawa algebra [G] in general, avoiding the "cheat" discussed at the end of Sect. 4.4.…”

Section: About 4-sequences and Cohomologymentioning

confidence: 99%

“…We try to explain the result of [17] in a nontrivial example, G being elementary abelian of order p 2 . Its main advantage over [12] is that it avoids taking the Iwasawa dual α(X S ) and addresses X S itself.…”

Section: About 4-sequences and Cohomologymentioning

confidence: 99%

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Fitting Ideals in Number Theory and Arithmetic

Greither

2021

Jahresber. Dtsch. Math. Ver.

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We describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.

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Section: About 4-sequences and Cohomologymentioning

confidence: 75%

“…Second Scenario [12,17] We adapt and simplify some notation from these papers for the present purpose.…”

Section: Example Scenariosmentioning

confidence: 99%

“…The second issue is treated in [17] completed by [19]; the issue of eliminating the major cheat is dealt with in [20]. We will discuss all this briefly in the final section.…”

Section: Example Scenariosmentioning

confidence: 99%

Section: About 4-sequences and Cohomologymentioning

confidence: 99%

Section: About 4-sequences and Cohomologymentioning

confidence: 99%

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Fitting Ideals in Number Theory and Arithmetic

Greither

2021

Jahresber. Dtsch. Math. Ver.

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Fitting ideals of p-ramified Iwasawa modules over totally real fields

Greither

Takenori

Kurihara

2021

Sel. Math. New Ser.

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We completely calculate the Fitting ideal of the classical p-ramified Iwasawa module for any abelian extension K/k of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results where we had to assume that only p-adic places may ramify in K/k. One of the important ingredients is the computation of some complexes in appropriate derived categories.

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Minimal resolutions of Iwasawa modules

Kataoka,

Kurihara

2024

Res. number theory

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In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian p-extension K/k of totally real fields and the cyclotomic $$\mathbb {Z}_p$$ Z p -extension $$K_{\infty }/K$$ K ∞ / K , we consider $$X_{K_{\infty },S}={{\,\textrm{Gal}\,}}(M_{K_{\infty },S}/K_{\infty })$$ X K ∞ , S = Gal ( M K ∞ , S / K ∞ ) where S is a finite set of places of k containing all ramifying places in $$K_{\infty }$$ K ∞ and archimedean places, and $$M_{K_{\infty },S}$$ M K ∞ , S is the maximal abelian pro-p-extension of $$K_{\infty }$$ K ∞ unramified outside S. We give lower and upper bounds of the minimal numbers of generators and of relations of $$X_{K_{\infty },S}$$ X K ∞ , S as a $$\mathbb {Z}_p[[{{\,\textrm{Gal}\,}}(K_{\infty }/k)]]$$ Z p [ [ Gal ( K ∞ / k ) ] ] -module, using the p-rank of $${{\,\textrm{Gal}\,}}(K/k)$$ Gal ( K / k ) . This result explains the complexity of $$X_{K_{\infty },S}$$ X K ∞ , S as a $$\mathbb {Z}_p[[{{\,\textrm{Gal}\,}}(K_{\infty }/k)]]$$ Z p [ [ Gal ( K ∞ / k ) ] ] -module when the p-rank of $${{\,\textrm{Gal}\,}}(K/k)$$ Gal ( K / k ) is large. Moreover, we prove an analogous theorem in the setting that K/k is non-abelian. We also study the Iwasawa adjoint of $$X_{K_{\infty },S}$$ X K ∞ , S , and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of $$X_{K_{\infty },S}$$ X K ∞ , S .

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Tate sequences and Fitting ideals of Iwasawa modules

Cited by 10 publications

References 10 publications

Fitting Ideals in Number Theory and Arithmetic

Fitting Ideals in Number Theory and Arithmetic

Fitting ideals of p-ramified Iwasawa modules over totally real fields

Minimal resolutions of Iwasawa modules

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