Ju l 1 99 4 ALL REGULATORS OF FLAT BUNDLES ARE TORSION 15 JUNE 1993

1.2 The theorem 1.1 contrasts the rank one case, where the secondary class of the flat bundle, associated to a representation ρ : π1(X) → GL1(C) = C ∗ is essentially the representation itself, so it may be completely arbitrary. But this freedom becomes limited when one looks at the line bundles over curves, corresponding via the Deligne-Ramakrishnan construction to elements of K2(X). The map r : K2(X) → H 1(X,C∗) which appears here, is called the Bloch-Beilinson regulator map. When X is defined over a number field, k, the image of K2(Xk) under r shows remarkable properties which were resumed in the celebrated Bloch-Beilinson conjecture [1], [20].