The additive dilogarithm
英文摘要:
We define an additive version of the Bloch group of a field, together with an additive dilogarithm and an Artin-Schreier realization.
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Spencer Bloch, Hélène Esnault
We define an additive version of the Bloch group of a field, together with an additive dilogarithm and an Artin-Schreier realization.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
Borel's construction of the regulator gives an injective map from the algebraic $K$-groups of a number field to its Deligne-Beilinson cohomology groups. This has many interesting arithmetic and geometric consequences. The formula for the regulator is expressed in terms of the classical polyogarithm functions. In this paper, we give a survey of the additive dilogarithm and the several different versions of the weight two regulator in the infinitesimal setting. We follow a historical approach which we hope will provide motivation for the definitions and the constructions.
We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This construction is similar to the construction of the norm map on Milnor $K$-theory. As an application, we express Chow dilogarithm in terms of Bloch-Wigner dilogarithm. Also, we obtain a new reciprocity law for four rational functions on an arbitrary algebraic surface with values in the pre-Bloch group.
For an algebraically closed field $k$ of characteristic 0, we give a cycle-theoretic description of the additive 4-term motivic exact sequence associated to the additive dilogarithm of J.-L. Cathelineau, that is the derivative of the Bloch-Wigner function, via the cubical additive higher Chow groups under one assumption. The 4-term functional equation of Cathelineau, an additive analogue of Abel's 5-term functional equation, is also discussed cycle-theoretically.
We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generatin…
Let $C_{2}$ be a smooth and projective curve over the ring of dual numbers of a field $k.$ Given non-zero rational functions $f,g,$ and $h$ on $C_{2},$ we define an invariant $ρ(f\wedge g \wedge h) \in k.$ This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture. Also using $ρ,$ we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park's construction in the case of cycles with modulus.
Using Totaro-Bloch-Kriz's linear fractional cycles Gangl and Muller-Stach recently prove the 5-term relations for the dilogarithm in Bloch's higher Chow group CH^2(F,3) and the Kummer-Spence relations in some group G(F) over an arbitrary field F where G(F) is isomorphic to CH^3(F,5) up to torsions under the Beilinson-Soule vanishing conjecture that CH^2(F,n)=0 for n>3. In this paper we show that Goncharov's 22-term relations for the trilogarithm also hold in G(F).
Our paper originated from a generalization of the Volume Conjecture to multisums of $q$-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; \cite{Ko}. We introduce the notion of a $q$-hypergeometric term (in short, $q$-term). The latter is a product of ratios of $q$-factorials in linear forms in several variables. In the first part of the paper, we show how to construct elements of the Bloch group (and its extended version) given a \qterm. Their image under the Bloch-Wigner map or the Rogers dilogarithm is a finite set of periods…
Bloch and Esnault defined additive higher Chow groups with modulus (m+1) on the level of zero cycles over a field k, denoted by TH^n(k,n;m), n,m >0. They prove that TH^n(k,n;1) is isomorphic to the group of absolute Kaehler differentials of degree n-1 over k. In this paper we generalize their result and show that TH^n(k,n;m) is isomorphic to W_mΩ^{n-1}_k, the group of degree n-1 elements in the generalized de Rham-Witt of length m over k. This complex was defined by Hesselholt and Madsen and generalizes the p-typical de Rham-Witt complex of Bloch-Deligne-Illusie. Before we can prove this theor…
There are two infinitesimal (i.e., additive) versions of the $K$-theory of a field $F$: one was introduced by Cathelineau, which is an $F$-module, and another one introduced by Bloch-Esnault, which is an $F^*$-module. Both versions are equipped with a regulator map, when $F$ is the field of complex numbers. In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group. Our results were motivated by two unrelated sources: Neumann's w…
We show how to make the additive Chow groups of Bloch-Esnault, Ruelling and Park into a graded module for Bloch's higher Chow groups, in the case of a smooth projective variety over a field. This yields a a projective bundle formula as well as a blow-up formula for the additive Chow groups of a smooth projective variety. In case the base-field admits resolution of singularieties, these properties allow us to apply the technique of Guille'n and Navarro Aznar to define the additive Chow groups "with log poles at infinity" for an arbitrary finite-type k-scheme X. This theory has the usual propert…
As an attempt to understand motives over $k[x]/(x^m)$, we define the cubical additive higher Chow groups with modulus for all dimensions extending the works of S. Bloch, H. Esnault and K. Rülling on 0-dimensional cycles. We give an explicit construction of regulator maps on the groups of 1-cycles with an aid of the residue theory of A. Parshin and V. Lomadze.
The higher Chow group with modulus was introduced by Binda-Saito as a common generalization of Bloch's higher Chow group and the additive higher Chow group. In this paper, we study invariance properties of the higher Chow group with modulus. First, we formulate and prove "cube invariance," which generalizes $\mathbb{A}^1$-homotopy invariance of Bloch's higher Chow group. Next, we introduce the nilpotent higher Chow group with modulus, as an analogue of the nilpotent algebraic $K$-group, and define a module structure on it over the big Witt ring of the base field. We deduce from the module stru…
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