Introduction to the Work of M. Kakde on the Non-commutative Main Conjectures for Totally Real Fields

Coates, John; Kim, Dohyeong

doi:10.1007/978-3-642-32199-3_1

Cited by 5 publications

(4 citation statements)

References 12 publications

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“…We briefly discuss some of the obstacles to generalising the approach of this article to prove higher dimension cases of the EIMC when a suitable µ = 0 hypothesis is not known. A serious obstacle is that a certain 'M H (G)-conjecture' is required to even formulate the higher dimension version of the EIMC, and that this is presently only known to hold under a suitable µ = 0 hypothesis (see [CK13,p. 5] and [CS12]).…”

Section: 5mentioning

confidence: 99%

Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

Johnston¹,

Nickel²

2018

American Journal of Mathematics

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Section: 5mentioning

confidence: 99%

Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

Johnston¹,

Nickel²

2018

American Journal of Mathematics

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“…We formulate the K-theoretic main conjecture (in analogy with the classical case [4]) as follows (we use the notation from Section 4).…”

Section: Relation To Algebraic K-theorymentioning

confidence: 99%

On the BDP Iwasawa main conjecture for modular forms

Lei

Zhao

2023

manuscripta math.

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Let p be a rational prime, and let X be a connected finite graph.In this article we study voltage covers X∞ of X attached to a voltage assignment α which takes values in some uniform p-adic Lie group G. We formulate and prove an Iwasawa main conjecture for the projective limit of the Picard groups Pic(Xn) of the intermediate voltage covers Xn, n ∈ N, and we prove one inclusion of a main conjecture for the projective limit of the Jacobians J(Xn).Moreover, we study the M H (G)-property of Zp G -modules and prove a necessary condition for this property which involves the µ-invariants of Zpsubcovers Y ⊆ X∞ of X. If the dimension of G is equal to 2, then this condition is also sufficient.

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“…The further reduction steps of previous approaches do not generalise easily as they rely on the µ = 0 hypothesis in a crucial way and hence presently there is no apparent way to deduce the EIMC for all admissible one-dimensional extensions without this hypothesis. Moreover, a serious obstacle to the case of admissible extensions of dimension greater than one is that in general a certain 'M H (G)-conjecture' is required to even formulate the EIMC in this situation, and that this is presently only known to hold under the µ = 0 hypothesis (see [CK13,p. 5] and [CS12]).…”

Section: Introductionmentioning

confidence: 99%

An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications

Johnston¹,

Nickel²

2020

Preprint

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Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for every admissible one-dimensional padic Lie extension whose Galois group has an abelian Sylow p-subgroup. Crucially, this result does not depend on the vanishing of any µ-invariant. As applications, we deduce the Coates-Sinnott conjecture away from its 2-primary part and new cases of the equivariant Tamagawa number conjecture for Tate motives.

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Introduction to the Work of M. Kakde on the Non-commutative Main Conjectures for Totally Real Fields

Cited by 5 publications

References 12 publications

Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

On the BDP Iwasawa main conjecture for modular forms

An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications

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