Iterated integrals and Epstein zeta functions with harmonic rational function coefficients (Q2640664)

In this paper the well-known transformation properties of theta-functions with (polynomial) harmonic coefficients as well as the associated Epstein zeta function are extended to theta functions with certain rational harmonic coefficients of low degree and a small number of variables. These theta functions are no longer modular forms, but the classical functional equation for the associated Epstein zeta function still holds. The proof of the various transformation laws are based on Chen's theory of iterated integrals. As an application this new Epstein zeta function divided by a product of two L-functions is used to generate for the Fermat quartic \(F_ 4:\) \(X_ 4+Y_ 4=1\) the Abel Jacobi image of the 1-cycle in \(Jac(F_ 4)\) given by \([F_ 4]-[\iota (F_ 4)]\).