By Don Blasius, Jonathan Rogawski (auth.), Alexander Reznikov, Norbert Schappacher (eds.)

This e-book is an outgrowth of the Workshop on "Regulators in research, Geom etry and quantity concept" held on the Edmund Landau middle for study in Mathematical research of The Hebrew college of Jerusalem in 1996. through the training and the conserving of the workshop we have been drastically helped through the director of the Landau middle: Lior Tsafriri throughout the time of the making plans of the convention, and Hershel Farkas in the course of the assembly itself. Organizing and operating this workshop used to be a real excitement, because of the professional technical aid supplied through the Landau heart regularly, and by means of its secretary Simcha Kojman specifically. we want to specific our hearty due to them all. despite the fact that, the articles assembled within the current quantity don't signify the complaints of this workshop; neither might all members to the ebook make it to the assembly, nor do the contributions herein inevitably mirror talks given in Jerusalem. within the creation, we define our view of the speculation to which this quantity intends to give a contribution. The an important goal of the current quantity is to compile ideas, equipment, and effects from research, differential in addition to algebraic geometry, and quantity concept with a purpose to paintings in the direction of a deeper and extra complete realizing of regulators and secondary invariants. Our thank you visit the entire individuals of the workshop and authors of this quantity. might the readers of this booklet get pleasure from and benefit from the mix of mathematical principles right here documented.

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**Regulators in Analysis, Geometry and Number Theory**

This booklet is an outgrowth of the Workshop on "Regulators in research, Geom etry and quantity conception" held on the Edmund Landau heart for study in Mathematical research of The Hebrew collage of Jerusalem in 1996. in the course of the practise and the conserving of the workshop we have been significantly helped through the director of the Landau heart: Lior Tsafriri through the time of the making plans of the convention, and Hershel Farkas throughout the assembly itself.

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Mark 1\vain, Adventures of Huckleberry Finn o Introduction 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 M of pure motives and then to try to construct a Hopf algebra or a co-Lie algebra H in the category M such that corepresentations of H in M give rise to (and conjecturally are equivalent to) mixed motives whose weight graded pieces lie in M.

Elliptic Motives 19 For example, viewing Q( I) as the alternating part of '}i®2, we have Z2(E 2 , n) [8J Q(1) = {F(x, y) E Z2(E 2, n) IF(x, y) = F(y, x) = -F(-x, y)} ® Q(I). ) We may define a complex N(E) in M(E) placing Za(E b, c) [8J '}i®b(-a) in degree 2a - b - c. 6) The differential is the usual alternating sum of face maps It follows from [4], Prop. 1), that the homology objects of this complex have the form CHa(E b, c) [8J '}i®b(-a), where CHa(E b, c) are the higher Chow groups [3]. 8) satisfying x .

Columns of (8) give resolutions of (AqTr* ®£)liO resp. 1. 1, we see that (A °Tr* ® £)liO and by the analogous argument also (A °70* ® £)81 is a resolution of £8' . 5. Assume we are given integrable subbundles T U, T S, TO, T' of T with T' = T U$ T S$ TO and such that T U$ TO and T S$ TO are integrable, but not necessarily T U$ rs. ;; A 27* ~ TU* ® TS*. ;; A27'* ~ TU* ® TS*. We thus get flat TU-resp. TS-connections on AqTs* resp. APTu* for all p, q 2: O. Given a vector bundle E on X with flat TU-resp.