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

2015/06/30 by Yajun Zhou
Mathematics · #math.NT #math.AG #msc:11F03 #msc:14E05

paper · pdf · doi:10.1007/s11139-016-9818-9

published as Ramanujan J. 42: 623-688 (2017); Erratum/addendum-ibid. 49:231-235 (2019) · i+46 pages. 1 table. Published version + erratum/addendum at the end of the article. A sequel to arXiv:1312.6352v4

arxiv created 2016/10/12 · arxiv updated 2019/04/24

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 Yⁿ=(1-X)^n-1X(1-αX) ( n∈\6,4,3,2\), we reformulate the Kontsevich-Zagier integral representations of weight-4 automorphic Green's functions G₂^\mathfrak H/\varGamma₀(N)(z₁,z₂) (N=4sin²(π/n )∈\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.

Citations

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