Syntomic regulators and values of $$p$$-adic $$L$$-functions. II. (Régulateurs syntomiques et valeurs de fonctions $$L$$ $$p$$-adiques. II.)(French)Zbl 0799.14010

[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.
Reviewer: M.Gros (Rennes)

MSC:

 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14F30 $$p$$-adic cohomology, crystalline cohomology

Zbl 0667.14006
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