A Nontrivial Algebraic Cycle in the Jacobian Variety of the Klein Quartic

Let X be a compact Riemann surface of genus g ≥ 2 and J(X) its Jacobian variety. By the Abel-Jacobi map X → J(X), X is embedded in J(X). The algebraic 1-cycle X −X− in J(X) is homologous to zero. Here we denote by X the image of X under the multiplication map by −1. If X is hyperelliptic, X = X in J(X). For the rest of this paper, suppose g ≥ 3. B. Harris [5] studied the problem whether the cycle X−X− in J(X) is algebraically equivalent to zero or not. The harmonic volume I for X was introduced by Harris [4], using Chen’s iterated integrals [2]. Let H denote the first integral homology group H1(X ;Z) of X . The harmonic volume I is defined to be a homomorphism (H) → R/Z. Here (H) is a certain subgroup of H. See Section 2 for the definition. Let ω be a third tensor product of holomorphic 1-forms on X . Suppose that ω + ω and (ω − ω)/ √ −1 belong to (H). If the cycle X −X− is algebraically equivalent to zero, then twice the values at both ω + ω and (ω − ω)/ √ −1 of the harmonic volume are zero modulo Z. Harris proved twice the value at ω+ ω of the harmonic volume for the Fermat quartic F (4) are nonzero modulo Z. This implies F (4)−F (4) is not algebraically equivalent to zero in J(F (4)) ([5], [6]). Ceresa [1] showed that X − X is not algebraically equivalent to zero for a generic X . We know few explicit nontrivial examples except for F (4). Let C denote the Klein quartic. See Section 4.1 for the definition. The aim of this paper is to show