Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians.

Let C be a curve of genus two, J(C) its Jacobian. The author defines a cycle Z(C, p, q) in the group H1(J(C),K2) = CH2(J(C), 1) which depends on the choice of two ordered Weierstrass points p and q. The cycle Z(C, p, q) can be interpreted as a degeneration of the Ceresa cycle Z(G) = G+−G− in the Jacobian of a curve of genus three. In a previous paper [ the author and G. P. Pirola studied the associated normal function overM3 and explicitly computed the infinitesimal invariant (G); if the curve G is not hyperelliptic, (G) determines the equation of the canonical image of the curve. In this paper the author studies the image of Z(C, p, q) under the regulator map and the infinitesimal invariant associated to the section of the family of primitive higher Jacobians obtained by varying the curve C in M2. He obtains a formula for the infinitesimal invariant (C) and shows that it determines the four undistinguished Weierstrass points. Using a monodromy argument he shows that the image of the group of indecomposable cycles H1(J(C),K2)/Pic(C) C under the regulator map is not finitely generated if the curve is (very) general. Finally the author studies a further degeneration to an elliptic curve E and defines a cycle Z(E) 2...in... H0(E,K2). Using infinitesimal methods he shows that the image of H0(E,K2) under the regulator map to H1(E,C) is not finitely generated.