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In the framework of Kontsevich-Zagier periods, we derive integralrepresentations for weight-$k$ automorphic Green's functions invariant undermodular transformations in $\varGamma_0(N)$ ($N\in\mathbb Z_{\geq1} $),provided that there are no cusp forms on the respective Hecke congruence groupswith an even integer weight $k\geq4$. These Kontsevich-Zagier integralrepresentations for automorphic Green's functions give explicit formulae forcertain Eichler-Shimura maps connecting Eichler cohomology to Maa{\ss} cuspforms. We construct integral representations for weight-4 Gross-Zagierrenormalized Green's functions (automorphic self-energy) from limit scenariosof the respective Kontsevich-Zagier integrals. We reduce the weight-4automorphic self-energy on $ X_0(4)(\mathbb C)=\varGamma_0(4)\setminus\mathfrakH^*$ to an explicit form, which supports an algebraicity conjecture of Grossand Zagier.

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