Beilinson’s Hodge Conjecture for K1 Revisited

the Betti cycle class map. If m = 0, then the Hodge conjecture (classical form) implies that clr,0 is surjective. Beilinson ([Be]) once conjectured that cl U r,m is always surjective. It was Jannsen ([J3]) who was the first to find a counterexample, in the case m = 1, where the complex numbers C are used in an essential way. In contrast to this, one expects the surjectivity of clr,m in the case where U is obtained via base extension from a variety defined over a number field. It turns out that Jannsen’s counterexample is indeed very special, being the complement of a closed subscheme of codimension r in a projective algebraic manifold X . In contrast to this, we show rather easily that the corresponding limit map