A formulation of conjectures on 𝑝-adic zeta functions in noncommutative Iwasawa theory

Fukaya, Takako; Kato, K.

doi:10.1090/trans2/219/01

Cited by 117 publications

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“…It is a special case of very general conjectures on motivic L-values put forward by Kato [14] and Kato and Fukaya [10]. Theorem 1.2 for l = 2 not only confirms Kato's point of view of L-values and l-adic L-functions as bases of determinant line bundles quite beautifully but also has other number theoretic consequences which were already noted in [9][5.1] and [6][Cor.…”

supporting

confidence: 68%

On the cyclotomic main conjecture for the prime 2

Flach¹

2011

Journal für die reine und angewandte Mathematik (Crelles Journa

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Let l be prime number, m 0 an integer prime to l andthe cyclotomic Iwasawa algebra of "tame level m 0 ". Usingétale cohomology one can define a certain perfect complex of Λ-modules ∆ ∞ (see section 1.2 below) and a certain basis L of the invertible In this article we give a proof of Theorem 1.2 for l = 2. This was claimed as Theorem 5.2 in the survey paper [9] but the proof given there, arguing separately for each height one prime q of Λ, is incomplete at primes q which contain l = 2. The argument given in [9][p.95] is an attempt to use only knowledge of the cohomology as well as perfectness of the complex ∆ ∞ q but it turns out that this information is insufficient. Here we shall use the techniques of the paper [6], such as the Coleman homomorphism and Leopoldt's result on the Galois structure of cyclotomic integer rings, to construct the complex ∆ ∞ q more explicitly and thereby verify Theorem 1.2.What is peculiar to the situation l = 2 is not only that Λ is never regular (due to the presence of the complex conjugation) but also that the l-adic L-function L is quite differently defined for even and odd characters. For even characters one interpolates first derivatives of Dirichlet L-functions via cyclotomic units, for odd characters one interpolates the values of these functions via Stickelberger elements. A proof of Theorem 1.2 therefore in some sense involves a "mod 2 congruence" between Stickelberger elements and cyclotomic units. Given the explicit nature of both objects it is perhaps not surprising that this congruence turns out to be an elementary statement which is arrived at, however, only after some rather arduous computations. The statement is the following. Let M ≡ 1 mod 4 be an integer,

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supporting

confidence: 68%

On the cyclotomic main conjecture for the prime 2

Flach¹

2011

Journal für die reine und angewandte Mathematik (Crelles Journa

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“…4(iv)]. These results are a converse to the result of Fukaya and Kato in [18] which asserts that, under suitable hypotheses, the 'non-commutative Tamagawa number conjecture' of loc. cit.…”

Section: Introductionmentioning

confidence: 61%

On descent theory and main conjectures in non-commutative Iwasawa theory

Burns¹,

Venjakob

2010

J. Inst. Math. Jussieu

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Abstract. We prove a 'Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of noncommutative Iwasawa algebras. We use these results to describe the precise connection between the main conjecture of non-commutative Iwasawa theory (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture.

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“…2 Apparently, the formula in §3.2.2 (7) of [Fukaya and Kato 2006] is not compatible with Deligne as claimed: Deligne identifies W ‫ޑ(‬ p ‫ޑ/‬ p ) via class field theory with ‫ޑ‬ × p by sending the geometric Frobenius automorphism to p, which induces, by a standard calculation applied to Definition (3.4.3.2) for epsilon constants of quasicharacters of ‫ޑ‬ × p in [Deligne 1973] (see for example [Hida 1993, §8.5 between (4a) and (4b)]), the formula (V χ , ψ) = χ(τ p ) −n σ ∈ n χ(σ ) −1 σ n , while in [Fukaya and Kato 2006] the factor is just χ(τ p ) n . Here χ : W ‫ޑ(‬ p ‫ޑ/‬ p ) → E × is a character which gives the action on the E-vector space V χ .…”

Section: Now Setmentioning

confidence: 99%

“…We choose the latter normalisation. Let S be the finite set of places of ‫ޑ‬ consisting of p, ∞ as well as the places of bad reduction of M. Now, according to the conjectures of [Fukaya and Kato 2006] there exists a ζ -isomorphism…”

Section: Global Functional Equationmentioning

confidence: 99%

“…In this appendix we recall some details of the formalism of determinant functors introduced in [Fukaya and Kato 2006] (see also [Venjakob 2007]). We fix an associative unital noetherian ring R. We write B(R) for the category of bounded complexes of (left) R-modules, C(R) for the category of bounded complexes of finitely generated (left) R-modules, P(R) for the category of finitely generated projective (left) R-modules and C p (R) for the category of bounded (cohomological) complexes of finitely generated projective (left) R-modules.…”

Section: Appendix B: Determinant Functorsmentioning

confidence: 99%

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Untitled

2013

Algebra Number Theory

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This paper contains a complete proof of Fukaya and Kato's -isomorphism conjecture for invertible -modules (the case of V = V 0 (r ), where V 0 is unramified of dimension 1). Our results rely heavily on Kato's proof, in an unpublished set of lecture notes, of (commutative) -isomorphisms for one-dimensional representations of G ‫ޑ‬ p , but apart from fixing some sign ambiguities in Kato's notes, we use the theory of (φ, )-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of Fukuya and Kato and extend it to the (slightly noncommutative) semiglobal setting. Finally we discuss some direct applications concerning the Iwasawa theory of CM elliptic curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves E over the extension of ‫ޑ‬ p which trivialises the p-power division points E( p) of E. In this sense the paper is complimentary to our work with Bouganis (Asian J. Math. 14:3 (2010), 385-416) on noncommutative Main Conjectures for CM elliptic curves.

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A formulation of conjectures on 𝑝-adic zeta functions in noncommutative Iwasawa theory

Cited by 117 publications

References 0 publications

On the cyclotomic main conjecture for the prime 2

On the cyclotomic main conjecture for the prime 2

On descent theory and main conjectures in non-commutative Iwasawa theory

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