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# Indecomposable Higher Chow cycles on Jacobians

## Abstract

We construct some natural indecomposable elements of $$CH^g(J(C),1)$$ with trivial regulator, and in particular, prove that  is uncountable for C a generic curve or a generic hyperelliptic curve of genus $$g \geq 3$$.
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... This does not contradicts Voisin's conjecture [42] on the countability of CH 2 ind (X, 1) Q , because CH 1 ind (X 2 , 1) Q = 0. It is known that CH p+1 ind (X, 1) Q is uncountable for certain varieties X, see [13]. The assumption on the nonvanishing of the image of ζ 2 by (4.1.2) is satisfied for some examples, see [14], [35]. ...
... It is known that CH p+1 ind (X, 1) Q is uncountable in some cases, see e.g. [13] for the case of the Jacobian of a curve, and also [14], [35] for examples such that the hypothesis of (0.3) is satisfied. (We can show that a higher cycle on a product of elliptic curves in [22] is decomposable by restricting to the generic fiber of the first projection.) ...
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