Comparison of Karoubi's regulator and the <i>p</i>-adic Borel regulator

Tamme, Georg

doi:10.1017/is011010026jkt165

Cited by 5 publications

(5 citation statements)

References 13 publications

(14 reference statements)

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“…Our motivation for developing the details of the p-adic regulator was similar to our motivation for [9], namely that the power series makes possible an algorithm for evaluating the p-adic valuation of the regulators on homology classes in the general linear group of number rings such as those given by the algorithm of [9]. In all other respects it should be clear to the reader that our approach has nothing to add to the more sophisticated methods of [22], [24] and [45].…”

Section: Introductionmentioning

confidence: 80%

“…where ν = 2s−1 i=0 x i Y i where the x i 's are the barycentric coordinates. The verification that this integral converges p-adically for a general 2s-tuple is quite delicate and is carried out in the Appendix to [45].…”

Section: Proofmentioning

confidence: 99%

“…Using an explicit p-adically analytic cocycle it is shown in [45] that the Karoubi-Hamida p-adic regulator, which equals R F by Theorem 4.5, coincides up to a non-zero rational factor with the Wagoner-Huber-Kings p-adic regulator of [47] and [22].…”

Section: Proofmentioning

confidence: 99%

“…In the course of working on [9] and [8] we noticed that our power series also converged p-adically, giving rise to an elementary, complete account of the regulator of [19] which culminates in R F of Corollary 4.3. Independently this construction was introduced in [45] and used to show that the regulators of [19] and [22] coincide up to a non-zero rational factor.…”

Section: Introductionmentioning

confidence: 99%

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p-adic cocycles and their regulator maps

Choo¹,

Snaith²

2010

J K-Theor

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Section: Introductionmentioning

confidence: 80%

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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p-adic cocycles and their regulator maps

Choo¹,

Snaith²

2010

J K-Theor

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“…e. m = 0 and d ≥ s. By [HK11] (see also [Ta12] Theorem 2.1) the Lazard isomorphism H s la (GL d (O), E) ≃ H s (gl d , E) (the subscript here stands for "locally analytic") is induced on the level of cochains by the map ∆ :…”

Section: Where Gl D (A) Acts Trivially On O(d) If (Gmentioning

confidence: 99%

Volume and symplectic structure for l-adic local systems

Pappas

2020

Preprint

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We introduce a notion of volume for an ℓ-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions can be viewed as arithmetic analogues of the volume and the Chern-Simons invariants of a representation of the fundamental group of a 3-manifold which fibers over the circle and of the symplectic form on the character varieties of a Riemann surface. We show that the absolute Galois group acts on the deformation space by conformal symplectomorphisms which extend to an ℓ-adic analytic flow. We also prove that the locus of the deformation space over which the local system suitably descends is the critical set of a collection of rigid analytic functions. The vanishing cycles of these functions give additional invariants. ContentsIntroduction 1 1. Preliminaries 6 2. K 2 invariants and 2-forms 10 3. Chern-Simons and volume 15 4. Representations of étale fundamental groups 25 5. Deformations and lifts 32 6. The symplectic nature of the Galois action 37 7. Appendix: Interpolation of iterates and flows 44 References 47 Notations: Throughout the paper N denotes the non-negative integers and ℓ is a prime. We denote by F ℓ the finite field of ℓ elements and by Z ℓ , resp. Q ℓ , the ℓ-adic integers, resp. ℓ-adic numbers. We fix an algebraic closure Qℓ of Q ℓ and we denote by | | ℓ (or simply | |), resp. v ℓ , the ℓ-adic absolute value, resp. ℓ-adic valuation on Qℓ , normalized so that |ℓ| ℓ = ℓ −1 , v ℓ (ℓ) = 1. We will denote by F a field of characteristic ℓ which is algebraic over the prime field F ℓ and by W (F), or simply W , the ring of Witt vectors with coefficients in F. If k is a field of characteristic = ℓ we fix an algebraic closure k. We denote by k sep the separable closure of k in k, by k(ζ ℓ ∞ ) = ∪ n≥1 k(ζ ℓ n ) the subfield of k sep generated over k by the (primitive) ℓ n -th roots of unity ζ ℓ n and by χ cycl : Gal(k sep /k) → Z × ℓ the cyclotomic character defined by σ(ζ ℓ n ) = ζ χ cycl (σ) ℓ nfor all n ≥ 1. We set G k = Gal(k sep /k). Finally, we will denote by ( ) * the Pontryagin dual, by ( ) ∨ the linear dual, and by ( ) × the units.

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Volume and symplectic structure for ℓ-adic local systems

Pappas

2021

Advances in Mathematics

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Comparison of Karoubi's regulator and the p-adic Borel regulator

Abstract: In this paper we prove the p-adic analogue of a result of Hamida [11], namely that the p-adic Borel regulator introduced by Huber and Kings for the Ktheory of a p-adic number field equals Karoubi's p-adic regulator up to an explicit rational factor.

Cited by 5 publications

References 13 publications

p-adic cocycles and their regulator maps

p-adic cocycles and their regulator maps

Volume and symplectic structure for l-adic local systems

Volume and symplectic structure for ℓ-adic local systems

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