Higher Chern classes in Iwasawa theory

Bleher, Frauke M.; Chinburg, Ted; Greenberg, Ralph; Kakde, Mahesh; Pappas, G.; Sharifi, Romyar T.; Taylor, Martin J.

doi:10.48550/arxiv.1512.00273

Cited by 5 publications

(21 citation statements)

References 40 publications

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“…This implies that (M +L/L)∩(N +L/L) = 0. (iii) By an exact sequence N/(p, S) → X K∞ /(p, S) → (X K∞ /N )/(p, S) → 0, we have N/(p, S) 1) . Therefore we have only to show N ∩ L ⊂ (p, S)N , since [N :…”

Section: Lemma 24mentioning

confidence: 98%

“…There is a conjecture that X K would be pseudo-null as a Z p [[Gal( K/K)]]module, which is called Greenberg's generalized conjecture ("pseudo-null" is defined in §3). Concerning this conjecture and its application, there are many studies (Bleher et al [1], Fujii [7], Itoh [11], Ozaki [16], and Minardi [14], etc.). However, even if Greenberg's generalized conjecture is true, it just states that the characteristic ideal of X K is trivial, so that it seems difficult to consider any analogues of the Iwasawa invariants and the Iwasawa main conjecture for X K .…”

Section: The Unramified Iwasawa Modulesmentioning

confidence: 99%

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Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

Miura¹,

Murakami²,

Otsuki³

et al. 2019

Preprint

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For a CM-field K and an odd prime number p, let K be a certain multiple Z pextension of K. In this paper, we study several basic properties of the unramified Iwasawa module X K of K as a Z p [[Gal( K /K)]]-module. Our first main result is a description of the order of a Galois coinvariant of X K in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic Z p -extension of K under a certain assumption on the splitting of primes above p. Second one is that if K is an imaginary quadratic field and p does not split in K, we give a necessary and sufficient condition for which X K is Z p [[Gal( K/K)]]-cyclic under several assumptions on the Iwasawa λ-invariant and the ideal class group of K, where K is the Z 2 p -extension of K.

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Section: Lemma 24mentioning

confidence: 98%

Section: The Unramified Iwasawa Modulesmentioning

confidence: 99%

Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

Miura¹,

Murakami²,

Otsuki³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…To study the codimension n support of a finitely generated Iwasawa module, we use the nth Chern class of its maximal codimension n submodule. This Chern class, as defined in [2], is the sum of the lengths of its localizations at the prime ideals of codimension n. For instance, the first Chern class of a finitely generated torsion Iwasawa module is the divisor defining its characteristic ideal.…”

Section: Introductionmentioning

confidence: 99%

“…For a finitely generated Iwasawa module M , we let t n (M ) denote the nth Chern class of the maximal submodule T n (M ) of M supported in codimension at least n. That is, t n (M ) is the formal sum t n (M ) = P length(T n (M ) P ) [P] over height n prime ideals P in the Iwasawa algebra. In the case that M = T n (M ), this is the nth Chern class c n (M ) of M considered in [2]. The invariant t 1 (M ) is naturally identified with the characteristic ideal of the torsion submodule of M , matching the classical definition.…”

Section: Introductionmentioning

confidence: 99%

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Exterior powers in Iwasawa theory

Bleher,

Chinburg,

Greenberg

et al. 2019

Preprint

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The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification.In this paper, we study more general and potentially much smaller Iwasawa modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of characteristic ideals of classical Iwasawa modules, hence by p-adic L-functions under the relevant CM main conjectures.

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Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Lei

Palvannan

2019

Forum of Mathematics, Sigma

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A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's 2-variable p-adic L-functions) and algebraic objects (two "everywhere unramified" Iwasawa modules) involving codimension two cycles in a 2-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field K (where an odd prime p splits) of an elliptic curve E, defined over Q, with good supersingular reduction at p. On the analytic side, we consider eight pairs of 2-variable p-adic L-functions in this setup (four of the 2-variable p-adic L-functions have been constructed by Loeffler and a fifth 2-variable p-adic L-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the Z 2 p -extension of K. We also provide numerical evidence, using algorithms of Pollack, towards a pseudo-nullity conjecture of Coates-Sujatha.1 The first subscript d of ρ d,n indicates the dimension of the Galois representation, while the second subscript n denotes a number one less than the Krull dimension of the ring R. In the settings we are interested in, the number n would denote the number of variables in the corresponding p-adic L-functions.2 The Panchishkin condition is a type of "ordinariness" assumption, introduced by Greenberg, while formulating the Iwasawa main conjecture for Galois deformations. See Section 4 in [14] for the precise definition. 2

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Higher Chern classes in Iwasawa theory

Cited by 5 publications

References 40 publications

Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

Exterior powers in Iwasawa theory

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

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