A Nontrivial Algebraic Cycle in the Jacobian Variety of the Fermat Sextic

B. Harris [5] defined the harmonic volume for the compact Riemann surface X of genus g ≥ 3, using Chen’s iterated integrals [2]. Let J(X) be the Jacobian variety of X. By the Abel-Jacobi map X → J(X), X is embedded in J(X). By a consideration of the special harmonic volume, Harris [6] proved that the algebraic cycle F (4)−F (4) is not algebraically equivalent to zero in J(F (4)). Here, F (4) is the Fermat quartic, which is a compact Riemann surface of genus 3. Ceresa [1] showed that the algebraic cycle X −X− is not algebraically equivalent to zero in J(X) for a generic X. We know few explicit nontrivial examples except for F (4). Harris [7] used the special feature of F (4) that its normalized period matrix has entries in a discrete subring of C. The Fermat sextic F (6) has the same feature. We use this and prove