<i>p</i>-adic cocycles and their regulator maps

Choo, Zacky; Snaith, Victor

doi:10.1017/is010008010jkt125

Cited by 8 publications

(8 citation statements)

References 24 publications

(50 reference statements)

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“…Proof. The identities in (1) and (2) and the cocycle identity in (4) are stated in Theorem 3.2 [CS11] for the evaluations at all classical points A → O F . For these evaluations, they follow from the expression for Φ s (T (X)) as a constant multiple of…”

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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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Section: Now Setmentioning

confidence: 99%

Volume and symplectic structure for l-adic local systems

Pappas

2020

Preprint

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“…In the meantime Hamida's cocycle was independently studied by Choo and Snaith [7] with the aim of deriving a formula for the regulator well suited for computer computations. Whereas they also obtain the continuity of Hamida's cocycle, for our comparison we need the stronger result of local analyticity.…”

Section: Introductionmentioning

confidence: 99%

“…The reader is encouraged to compare this with the method of Choo and Snaith [7] based on nice explicit computations of integrals.…”

Section: Introductionmentioning

confidence: 99%

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

Tamme¹

2011

J. K-Theory

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“…Varios agentes como los metales, luz ultravioleta, temperaturas extremas y otros contaminantes están asociados con la sobreproducción de especies tóxicas de oxígeno y daño celular en los productores primarios. 1,2 Durante el metabolismo celular, una pequeña proporción (2-3%) de las especies reactivas de oxígeno puede escapar del rol protector de los mecanismos antioxidantes, causando daño oxidativo a los componentes celulares. El desequilibrio entre la producción y la neutralización de las especies reactivas de oxígeno por los mecanismos antioxidantes se llama estrés oxidativo.…”

Section: Introductionunclassified

Daño oxidativo en la microalga Pseudokirchneriella subcapitata expuesta a aguas receptoras de un efluente minero en del Río Blanco (V Región, Chile)

Aránguiz¹,

Gaete²,

Hidalgo³

et al. 2009

Quím. Nova

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Recebido em 5/3/09; aceito em 19/5/09; publicado na web em 20/10/09 OXIDATIVE DEMAGE IN THE MICROALGAE Pseudokirchneriella subcapitata EXPOSED TO RECEIVING WATERS OF A MINING EFFLUENT IN THE RIO BLANCO (V REGION, CHILE). In this investigation antioxidant response and toxicity of metals in receiving water effluent miner in the Blanco river in Pseudokirchneriella subcapitata was assessed. The catalase activity, lipid damage through Tbars, the growth rate of was determined. The result showed an inhibition of the growth rate of P. subcapitata which correlated with increased catalase activity and the lipid liperoxidation. These responses were correlated with the concentrations of copper and iron.

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p-adic cocycles and their regulator maps

Abstract: We derive a power series formula for the p-adic regulator on the higher dimensional algebraic K-groups of number fields. This formula is designed to be well suited to computer calculations and to reduction modulo powers of p.

Cited by 8 publications

References 24 publications

Volume and symplectic structure for l-adic local systems

Volume and symplectic structure for l-adic local systems

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

Daño oxidativo en la microalga Pseudokirchneriella subcapitata expuesta a aguas receptoras de un efluente minero en del Río Blanco (V Región, Chile)

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