On the harmonic volume of Fermat curves

We prove that B. Harris’ harmonic volume of the Fermat curve of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of infinite order if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a prime divisor greater than 7. The statement is equivalent to the statement that the Griffiths’ Abel-Jacobi image of the Ceresa cycle of such a curve is of infinite order for every choice of base point. In particular, these cycles are of infinite order modulo rational equivalence.