Kontsevich-Zagier Integrals for Automorphic Green's Functions. I

In the framework of Kontsevich-Zagier periods, we derive integral representations for weight-$k$ automorphic Green’s functions invariant under modular transformations in ${\Gamma }_0(N)$ ($N\in {\mathbb{Z}}_{\ge 1}$), provided that there are no cusp forms on the respective Hecke congruence groups with an even integer weight $k\ge 4$. These Kontsevich-Zagier integral representations for automorphic Green’s functions give explicit formulae for certain Eichler-Shimura maps connecting Eichler cohomology to Maa{\ss} cusp forms. We construct integral representations for weight-4 Gross-Zagier renormalized Green’s functions (automorphic self-energy) from limit scenarios of the respective Kontsevich-Zagier integrals. We reduce the weight-4 automorphic self-energy on $ X_0(4)(\mathbb C)=\varGamma_0(4)\setminus\mathfrak H^*$ to an explicit form, which supports an algebraicity conjecture of Gross and Zagier.