Dirichlet motives via modular curves

Generalizing ideas of Anderson, Harder has proposed a construction of exten sions of Tate-motives (more precisely of Hedge structures and Galois module s, respectively) in terms of modular curves. The aim of this paper is to co nstruct directly those elements of motivic cohomology of SpecQ(mu(N)) (i.e. in K-* (SpecQ(mu(N)))) which induce these extensions in absolute Hedge coh omology and continuous Galois cohomology. We give two such constructions an d prove that they are equivalent. The key ingredient is Beilinson's Eisenst ein symbol in motivic cohomology of powers of the universal elliptic curve over the modular curve. We also compute explicitly the Harder-Anderson elem ent in absolute Hedge cohomology. Tt is given in terms of Dirichlet-L-funct ions. As a corollary, we get a new proof of Beilinson's conjecture for Diri chlet-L-functions. A second paper [HuK] treats the explicit computation in the l-adic case. (C) Elsevier, Paris.