Iwasawa Theory for One-Parameter Families of Motives

Barth, Peter

doi:10.1142/s1793042112501357

Cited by 2 publications

(2 citation statements)

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“…We formulate a Main conjecture for Selmer groups of Ad 0 (ρ 0 ) along the irreducible component I. A Main conjecture for Galois representations attached to the ordinary Hida family was also formulated by Barth in his thesis [Bar09]. Our formulation is slightly different from his formulation.…”

Section: Introductionmentioning

confidence: 92%

Noncommutative Iwasawa theory arising from Hecke algebras

Aribam¹

2016

Preprint

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Let p be an odd prime and f be a nearly ordinary Hilbert modular Hecke eigenform defined over a totally real field F . Let I be an irreducible component of the universal nearly ordinary or locally cyclotomic deformation of the representation of Gal F that is associated to f . We study the deformation rings over a p-adic Lie extension F ∞ that contains the cyclotomic Z p -extension of F . More precisely, we prove a control theorem about these rings. We introduce a category M I H (G), where G = Gal(F ∞ /F ) and H = Gal(F ∞ /F cyc ), which is the category of modules which are torsion with respect to a certain Ore set, which generalizes the Ore set introduced by Venjakob. For Selmer groups which are in this category, we formulate a Main conjecture in the spirit of Noncommutative Iwasawa theory. We then set up a strategy to prove the conjecture by generalizing work of Burns, Kato, Kakde, and Ritter and Weiss. This requires appropriate generalizations of results of Oliver and Taylor, and Oliver on Logarithms of certain K-groups, which we have presented here.

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Section: Introductionmentioning

confidence: 92%

Noncommutative Iwasawa theory arising from Hecke algebras

Aribam¹

2016

Preprint

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“…Analogous results have been obtained for Selmer groups of Galois representations over the anticyclotomic -extension of an imaginary quadratic base field K (see [HL19]) and for signed Selmer groups of Galois representations over the cyclotomic -extension of a number field in the non-ordinary setting (see, e.g., [Pon20, Section 3]). Moreover, there exist vast generalisations to Selmer groups attached to families of modular forms (see, e.g., [EPW06, Sha09, Bar13]).…”

Section: Introductionmentioning

confidence: 99%

Fine Selmer groups of congruent p-adic Galois representations

Kleine

Müller²

2021

Can. Math. Bull.

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We compare the Pontryagin duals of fine Selmer groups of two congruent -adic Galois representations over admissible pro-, -adic Lie extensions ∞ of number fields . We prove that in several natural settings the -primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the -invariants. In the special case of a Z -extension ∞ / , we also compare the Iwasawa -invariants of the fine Selmer groups, even in situations where the -invariants are non-zero. Finally, we prove similar results for certain abelian non--extensions.

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Iwasawa Theory for One-Parameter Families of Motives

Cited by 2 publications

References 19 publications

Noncommutative Iwasawa theory arising from Hecke algebras

Noncommutative Iwasawa theory arising from Hecke algebras

Fine Selmer groups of congruent p-adic Galois representations

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