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arXiv · 1506.00318

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

Yajun Zhou
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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-αX)$ ($ 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)$ ($N=4\sin^2(π/n )\in\{1,2,3,4\}$), 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.

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Yajun Zhou. 2016-10-12. Kontsevich-Zagier Integrals for Automorphic Green's Functions. II. https://doi.org/10.1007/s11139-016-9818-9%2010.1007%2Fs11139-018-0100-1

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