Euler system for Galois deformations

Ochiai, Tadashi

doi:10.5802/aif.2091

Cited by 33 publications

(81 citation statements)

References 18 publications

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“…Our approach is altogether different from theirs: Instead of interpolating Euler systems (as in [Och05,How07]), we instead deform Kolyvagin systems. We remark that a Kolyvagin system has exactly the same use as an Euler system, when they are used to bound Selmer groups.…”

Section: The Resultsmentioning

confidence: 99%

“…Then Serre's conjecture [Ser87] (as proved in [KW09,Kis09a]) implies that ρ arises from an ordinary newform. Hida associates in [Hid86b,Hid86a] such f a family of ordinary modular forms and a Galois representation T attached to the family, with coefficients in the universal ordinary Hecke algebra H. Thanks to the "R = T " theorems proved in [Wil95,TW95] (and their refinements) it follows that H is the universal ordinary deformation ring of ρ, parametrizing all ordinary deformations of ρ. Ochiai in [Och05] (resp., Howard in [How07]) has studied the Iwasawa theory of this family of Galois representations by interpolating Kato's Euler system (resp., Heegner points) for each member of the family to a 'big' Euler system for the whole p-ordinary family.…”

Section: When (Hnob) Holds True Mazur Proved That R(ρ)mentioning

confidence: 99%

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Deformations of Kolyvagin systems

Büyükboduk

2016

Ann. Math. Québec

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ABSTRACT. Ochiai has previously proved that the Beilinson-Kato Euler systems for modular forms interpolate in nearly-ordinary p-adic families (Howard has obtained a similar result for Heegner points), based on which he was able to prove a half of the two-variable main conjectures. The principal goal of this article is to generalize Ochiai's work in the level of Kolyvagin systems so as to prove that Kolyvagin systems associated to Beilinson-Kato elements interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus) and use what we call universal Kolyvagin systems to attempt a main conjecture over the eigencurve. Along the way, we utilize these objects in order to define a quasicoherent sheaf on the eigencurve that behaves like a p-adic L-function (in a certain sense of the word, in 3-variables).

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Section: The Resultsmentioning

confidence: 99%

Section: When (Hnob) Holds True Mazur Proved That R(ρ)mentioning

confidence: 99%

Deformations of Kolyvagin systems

Büyükboduk

2016

Ann. Math. Québec

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“…To this aim, we prove some preliminary results from [14] over general normal domains by making necessary modifications. The next lemma is necessary for a technical reason and it will be shown in Lemma 8.4 how to use it in a more concrete situation.…”

Section: R-modules To Torsion R W(f)mentioning

confidence: 99%

“…(3) For all but finitely many height-one primes: (F) ( Main Theorem B will be crucial in a forthcoming paper [15], where we plan to compare characteristic ideals of certain torsion modules arising from Iwasawa theory as developed in [14] (for example, those torsion modules arise as the Pontryagin dual modules of Selmer groups associated with two-dimensional certain Galois representations with values in a complete local ring with finite residue field).…”

Section: ) As Followsmentioning

confidence: 99%

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Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

Ochiai¹,

Shimomoto²

2015

Nagoya Math. J.

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ABSTRACT. In this article, we prove a strong version of local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen-Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

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Big Heegner point Kolyvagin system for a family of modular forms

Büyükboduk

2013

Sel. Math. New Ser.

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The principal goal of this paper is to develop Kolyvagin's descent to apply with the big Heegner point Euler system constructed by Howard for the big Galois representation $\mathbb{T}$ attached to a Hida family $\mathbb{F}$ of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family $\mathbb{F}$ at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.Comment: 24 pages. Many updates to previous version. Added Remark 3.2 explaining in detail how to deduce an exact sequence of the form (3.1) verifying (3.2

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Euler system for Galois deformations

Cited by 33 publications

References 18 publications

Deformations of Kolyvagin systems

Deformations of Kolyvagin systems

Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

Big Heegner point Kolyvagin system for a family of modular forms

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