Syntomic regulators and values of \(p\)-adic \(L\)-functions. II
type: Item
instance of: scholarly article
title: Syntomic regulators and values of \(p\)-adic \(L\)-functions. II
zbMATH Open document ID: 0799.14010
DOI: 10.1007/BF01231754
author: Michel Gros
published in: Inventiones Mathematicae
publication date: 1994-11-17T00:00:00Z
full work available at URL: https://eudml.org/doc/144163
review text: [For part I see Invent. Math. 99, 292--320 (1990; Zbl 0667.14006).] Let \(k\) be a finite field and \(W\) the ring of Witt vectors on \(k\). In part I (loc. cit.) we have constructed (for \(X\) a flat quasi-projective scheme over \(W)\) \(p\)-adic regulators \[ c_{i,j} : K_ j (X) \to H^{2i-j} (X,s_ \infty (i)_ X) \] by mimicking Beilinson's construction of higher regulators but using some ``syntomic'' sheaves \(s_ \infty (i)_ X\) introduces in their studies of Hodge-Tate conjecture by Fontaine-Messing in place of Deligne's sheaves \(\mathbb{Z} (i)\) used by Beilinson. The paper gives the computation of the image by these \(p\)-adic regulators, more precisely by \(c_{i,2i - 1}\) \((i \geq 2)\) of some famous elements constructed by Beilinson in \(K_{2i - 1} (W) \otimes \mathbb{Q}\). The construction of these elements is usually done by using relative \(K\)-theory of affine schemes and it is necessary, for computational purpose, to factorize our construction of \(p\)-adic regulators by rigid regulators using rigid variants \(s(i)_{X,rig}\) of the sheaves \(s_ \infty (i)_ X\). These sheaves \(s_{rig} (i)_ X\) have better homotopy properties and the rigid point of view (or Monsky- Washnitzer point of view of overconvergent algebras) can be viewed here as analog of the use of De Rham complexes with log poles by Beilinson. We define these sheaves \(s(i)_{X,rig}\) and give the construction of the refined \(p\)-adic regulators (it must be noticed that p. 69, 1.7 is not sufficient as given and has to be precised). After doing this, the computation follows faithfully the classical case and the final result involves values of \(p\)-adic polylogarithms. Some applications in direction of special values of \(p\)-adic \(L\) functions of Dirichlet characters are given as well as comparisons with Deligne-Soulé elements in \(K\)-theory.
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 549839
zbMATH Keywords: \(p\)-adic \(L\)-functions
zbMATH Keywords: syntomic cohomology
zbMATH Keywords: Monsky-Washnitzer cohomology
zbMATH Keywords: \(p\)-adic regulators
MaRDI profile type: Publication
cites work: Q3972724
cites work: Dilogarithms, regulators and \(p\)-adic \(L\)-functions
cites work: Riemann-Roch theorems for higher algebraic K-theory
cites work: Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique
cites work: Syntomic regulators and values of \(p\)-adic \(L\)-functions. I. Appendix by Masato Kurihara
cites work: Formal cohomology. I
cites work: Formal cohomology. II: The cohomology sequence of a pair
cites work: K-théorie des anneaux d'entiers de corps de nombres et cohomologie etale
cites work: Q3744286
OpenAlex ID: W2320006757
description: scientific article; zbMATH DE number 549839