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## Über dieses Buch

This book is an outgrowth of the Workshop on "Regulators in Analysis, Geom­ etry and Number Theory" held at the Edmund Landau Center for Research in Mathematical Analysis of The Hebrew University of Jerusalem in 1996. During the preparation and the holding of the workshop we were greatly helped by the director of the Landau Center: Lior Tsafriri during the time of the planning of the conference, and Hershel Farkas during the meeting itself. Organizing and running this workshop was a true pleasure, thanks to the expert technical help provided by the Landau Center in general, and by its secretary Simcha Kojman in particular. We would like to express our hearty thanks to all of them. However, the articles assembled in the present volume do not represent the proceedings of this workshop; neither could all contributors to the book make it to the meeting, nor do the contributions herein necessarily reflect talks given in Jerusalem. In the introduction, we outline our view of the theory to which this volume intends to contribute. The crucial objective of the present volume is to bring together concepts, methods, and results from analysis, differential as well as algebraic geometry, and number theory in order to work towards a deeper and more comprehensive understanding of regulators and secondary invariants. Our thanks go to all the participants of the workshop and authors of this volume. May the readers of this book enjoy and profit from the combination of mathematical ideas here documented.

## Inhaltsverzeichnis

### Cohomology of Congruence Subgroups of SU (2, 1) p and Hodge Cycles on Some Special Complex Hyperbolic Surfaces

Abstract
Let G be a semisimple algebraic group over a number field F and set G = ∏ vS Gv, where S is the set of archimedean places of F. As is well-known, the cohomology of a cocompact lattice Γ ⊂ G is expressed in terms of the decomposition
$${L^2}\left( {\Gamma \backslash {G_\infty }} \right) \simeq \hat \oplus m(\pi ,\Gamma )\pi$$
Don Blasius, Jonathan Rogawski

### Remarks on Elliptic Motives

Abstract
This paper grew out of conversations at the conference with A. Goncharov and A. Levin on their paper [7]. I want to offer an interpretation of their results along lines developed in [4]. The idea, which I learned from A. Beilinson and P. Deligne [5], is to assume one is given some category Μ of pure motives and then to try to construct a Hopf algebra or a co-Lie algebra H in the category Μ such that corepresentations of H in Μ give rise to (and conjecturally are equivalent to) mixed motives whose weight graded pieces lie in Μ. The focus becomes the study of H and its representations just as in number theory one studies the Galois group and its representations. In the case of motives, H is constructed using algebraic cycles, and cycle classes in some Weil cohomology lead to realizations of the mixed motives.
Spencer Bloch

### On Dynamical Systems and Their Possible Significance for Arithmetic Geometry

Abstract
In the papers [D1], [D5], [D3] a cohomological formalism for algebraic schemes X 0 over spec ℤ or spec ℚ was conjectured which would explain many of the expected properties of motivic L-series. All consequences of this very rigid formalism that I could imagine turned out to be either provable [D2], [D4], [DN], [Sa] or to amount to some well known conjectures on L-series of motives, as for example the Riemann hypotheses and the Artin conjecture—both generalized to the context of motives—and the Bloch Beilinson conjectures on vanishing orders.
Christopher Deninger

### Algebraic Differential Characters

Abstract
In [5], Cheeger and Simons defined on a C manifold X a group of differential characters Ĥ 2n (X, ℝ/ℤ), which is an extension of the global ℝ-valued closed forms of degree 2n having ℤ periods, by the group H 2n-1 (X, ℝ/ℤ). (In fact, they write Ĥ 2n-1 (X, ℝ/ℤ), but the notation 2n rather than (2n-1) fits better with weights in algebraic geometry). Similarly, there is a group of (complex) differential characters Ĥ 2n (X,ℂ/ℤ), presented as an extension of the global ℂ-valued closed forms of degree 2n having ℤ periods, by the group Ĥ 2n-1 (X,ℂ/ℤ). The group Ĥ 2n (X, ℝ/ℤ) (resp. Ĥ 2n (X,ℂ/ℤ) is also presented as an extension of the Betti cohomology group Ĥ 2n (X, ℤ) by global ℝ-valued (resp. ℂ-valued) differential forms of degree 2n - 1, modulo the closed ones with ℤ periods.
Hélène Esnault

### Some Computations in Weight 4 Motivic Complexes

Abstract
We perform computations using the candidates for motivic cohomology as proposed by Goncharov, thereby giving evidence for his general conjecture about motivic complexes which would imply (among others) Zagier’s conjecture on polylogarithms and values of Dedekind zeta functions. As a by-product, we obtain a family of functional equations for the 4-logarithm.
Herbert Gangl

### Geometry of the Trilogarithm and the Motivic Lie Algebra of a Field

Abstract
We express explicitly the Aomoto trilogarithm by classical trilogarithms and investigate the algebro-geometric structures lying behind it: different realizations of the weight three motivic complexes. Applying these results, we describe the motivic structure of the Grassmannian tetralogarithm function and determine the structure of the motivic Lie coalgebra in degrees ≤ 4. Using this we give an explicit construction of the Bore1 regulator map
$$r4\;:{K_7}\left( \mathbb{C} \right) \to \mathbb{R}$$
which together with Borel’s theorem leads to results about ζ F (4).
A. B. Goncharov

### Complex Analytic Torsion Forms for Torus Fibrations and Moduli Spaces

Abstract
We construct analytic torsion forms for line bundles on holomorphic fibrations by tori, which are not necessarily Kähler fibrations. This is done by double transgressing the top Chern class. The forms are given in terms of Epstein zeta functions. Also, we establish a corresponding double transgression formula and an anomaly formula. The forms are investigated more closely for the universal bundle over the moduli space of polarized abelian varieties and for the bundle of Jacobians over the Teichmüller space.
Kai Köhler

### Théorèmes de Lefschetz et de Hodge arithmétiques pour les variétés admettant une décomposition cellulaire

Abstract
Soit π : X → Spec ℤ une variété arithmétique (i.e., un schéma plat, projectif, intègre et régulier sur Spec ℤ) de dimension absolue n + 1, et notons $$\widehat {C{H^*}}{\left( X \right)_\mathbb{R}}$$ l’anneau de Chow arithmétique réel défini dans [GS3]; c’est un anneau gradué muni d’une application degré:
$$\widehat {\deg }:{\widehat {CH}^{n + 1}}{\left( X \right)_\mathbb{R}} \to \mathbb{R}$$
Klaus Künnemann, Vincent Maillot

### Polylogarithmic Currents on Abelian Varieties

Abstract
We construct a certain collection of currents on a universal family of abelian varieties. The cohomology classes of these currents are rational; so our currents are a natural generalization of the Eisenstein series on the modular curve.
Andrey Levin

### Secondary Analytic Indices

Abstract
We define real-valued characteristic classes of flat complex vector bundles, and flat real vector bundles with a duality structure. We construct pushforwards of such vector bundles with vanishing characteristic classes. These pushforwards involve the analytic torsion form in the first case and the eta form of the signature operator in the second case. We show that the pushforwards are independent of the geometric choices made in the constructions and hence are topological in nature. We give evidence that in the first case, the pushforwards are given topologically by the Becker-Gottlieb-Dold transfer.
John Lott

### Variations of Hodge—de Rham Structure and Elliptic Modular Units

Abstract
In this paper, we give a conceptual interpretation of generalized elliptic units in terms of variations of Hodge-de Rham structure, and of (elliptic) polylogarithms.
Jörg Wildeshaus

### Backmatter

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