  In the framework of Kontsevich-Zagier periods, we derive integral
representations for weight-$k$ automorphic Green's functions invariant under
modular transformations in $\varGamma_0(N)$ ($N\in\mathbb Z_{\geq1} $),
provided that there are no cusp forms on the respective Hecke congruence groups
with an even integer weight $k\geq4$. 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.
