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

Abstract

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 Maaß 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 $X_0(4)(\mathbb{C})={\Gamma }_0(4)\setminus {\mathfrak{H}}^{\ast }$ to an explicit form, which supports an algebraicity conjecture of Gross and Zagier.

i+69 pages, 1 figure. Minor clarifications and notational changes. Numerical supplement (NumSuppAGF1.nb) downloadable from http://arxiv.org/format/1312.6352

References in corpus (1)

• Elliptic integral evaluations of Bessel moments

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