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

TL;DR

This paper introduces interaction entropies linked to algebraic curves to reformulate integrals of automorphic Green's functions, revealing algebraic relations between automorphic self-energies and Green's function values.

Contribution

It develops a geometric framework using interaction entropies for automorphic Green's functions, connecting algebraic curves with automorphic forms and Green's function relations.

Findings

Reformulation of Kontsevich-Zagier integrals using geometric entropies

Establishment of algebraic relations between automorphic self-energies and Green's functions

Introduction of interaction entropies for Legendre-Ramanujan curves

Abstract

We introduce interaction entropies, which can be represented as logarithmic couplings of certain cycles on a class of algebraic curves of arithmetic interest. In particular, via interaction entropies for Legendre-Ramanujan curves $ Y^n=(1-X)^{n-1}X(1-\alpha X)$ ($ n\in\{6,4,3,2\}$), we reformulate the Kontsevich-Zagier integral representations of weight-4 automorphic Green's functions $ G_2^{\mathfrak H/\overline{\varGamma}_0(N)}(z_1,z_2)$ ($N=4\sin^2(\pi/n )\in\{1,2,3,4\}$), in a geometric context. These geometric entropies allow us to establish algebraic relations between certain weight-4 automorphic self-energies and special values of weight-6 automorphic Green's functions.